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APPLICATIONS OF RENORMALIZED COUPLED-CLUSTER
METHODS TO MOLECULAR POTENTIAL ENERGY
SURFACES, SPECTROSCOPY, AND REACTION
MECHANISMS

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Michael J. McGuire

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APPLICATIONS OF RENORMALIZED COUPLED-CLUSTER
METHODS TO MOLECULAR POTENTIAL ENERGY
SURFACES, SPECTROSCOPY, AND REACTION MECHANISMS
By

Michael J. McGuire

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

2007

ABSTRACT
APPLICATIONS OF RENORMALIZED COUPLED-CLUSTER
METHODS TO MOLECULAR POTENTIAL ENERGY
SURFACES, SPECTROSCOPY, AND REACTION MECHANISMS
By

Michael J. McGuire

Most single-reference ab initio methods can produce good results for closed shell
molecules near equilibrium geometries. However, these methods fail when bond dis-
tances are stretched and bonds are broken and when the molecule possesses a signif-
icant amount of diradical character. This results in a drastic decrease in the quality
of molecular potential energy surfaces (PESS) along bond breaking coordinates, as
well as in calculations involving diradical species. To rectify this situation one would
normally have to rely on multi-reference methods, which can be difficult to set-up and
very expensive. However, the single-reference renormalized coupled-cluster methods,
which are approximations to the more general method of moments of coupled-cluster
equations, have been shown to give excellent results for PESS involving the breaking
of single bonds at relatively low cost. This study focuses on extending the use of
the completely-renormalized (CR) CCSD(T) method to entire reactive ground-state
PESs using the Be + HF reaction as an example, and the use of the CR-CCSD(T)
method in systems which display diradical character, such as the trans isomer of
imine peroxide (HNOO), the Cope rearrangement of 1,5-hexadiene, and the Bergman

cyclization of enediyne molecules.

The CR-CCSD(T) results are compared to the results of standard CC calculations, as

well as those obtained in multi-reference and density functional theory calculations.

ACKNOWLEDGMENTS

I would like to express my thanks to the following people for their part in this
work and in my graduate studies:
my advisor, Dr. Piotr Piecuch;
my advisorial committee, Dr. Katharine Hunt, Dr. Marcos Dantus, Dr. James Jack-
son, and Dr. James McCusker;
Dr. Karol Kowalski;
The Piecuch group: Dr. Marta Wloch, Dr. Armagan Kinal, Maricris, and Jeff;
My friends: DJ, Paul, and everyone else;

And finally my family.

iv

TABLE OF CONTENTS

List of Tables ............................................................................................................ vii
List of Figures ........................................................................................................... ix
1. Introduction ........................................................................................................... 1
2. Project Objectives ................................................................................................. 8

3. Theory: Method of Moments of Coupled-Cluster Equations and the
Renormalized Coupled-Cluster Approaches ........................................................... 9
3.1. The Exact MMCC Formalism ................................................................... 9

3.2. The Renormalized and Completely Renormalized CCSD(T)
and CCSD(TQf) Methods ....................................................................... 14

4. Applications: Molecular Potential Energy Surfaces for the Be + HF

Reaction ............................................................................................................... 22
4.1. Background Information and Motivation ................................................ 22
4.2. Computational Details ............................................................................ 23

4.3. Results of the Standard, Renormalized, and Completely

Renormalized CCSD(T) Calculations Employing the MIDI
Basis Set .................................................................................................. 25

4.4. Results of the Standard, Renormalized, and Completely
Renormalized CCSD(T) Calculations Employing the
cc-pVTZ and cc-pVQZ Basis Sets ........................................................... 32

4.5. Conclusion ............................................................................................... 41

5. Applications: The Vibrational Spectrum of trans-Imine Peroxide
(HNOO) ............................................................................................................... 42

5.1. Background Information and Motivation ................................................ 42
5.2. Computational Details ............................................................................ 44

5.3. Standard and Completely Renormalized CCSD(T) and
CCSDéTQ Calculations of the Geometry, Anharmonic
Force iel , and Vibrational Spectrum of trans-HNOO ......................... 48

5.4. Conclusion ............................................................................................... 53

6. Applications: The Mechanisms of the Cope Rearrangement of

1,5-Hexadiene and the Bergman Cyclizations of Enediynes ................................. 55
6.1. Background Information and Motivation ................................................ 55
6.2. Computational Details ............................................................................ 59

6.3. Results for the Mechanism of the Cope Rearrangement of

1,5—Hexadiene .......................................................................................... 63

6.4. Results for the Bergman Cyclization of Enediynes .................................. 72

6.5. Conclusion ............................................................................................... 78

7. Summary and Concluding Remarks ..................................................................... 80

Appendix A. The PESs for the BeFH system as described by the
MIDI basis set .................................................................................... 7 8

References ............................................................................................................... .139

vi

List of Tables

Table 1. Maximum absolute errors, relative to full CI, in the CCSD, CCSD(T),
R—CCSD(T), and CR-CCSD(T) results for the ground-state PES of the
collinear BeFH system (in millihartree), as obtained with the MIDI basis set.
The RBe_p and 1211.}: values are in bohr .................................................................. 93

Table 2. Maximum absolute errors (in eV), relative to MRCI(Q), in the CCSD,
CCSD(T) and CR—CCSD(T) energies for the ground-state PES of the BeFH
system, as described by the cc-pVTZ and cc-pVQZ basis sets, at a Be—F—H
angle 6’ of 180°. The R394: and R34: values are in bohr. ....................................... 94

Table 3. Maximum absolute errors (in eV), relative to MRCI(Q), in the CCSD,
CCSD(T) and CR-CCSD(T) energies for the ground-state PES of the BeFH
system at Be—F-H angles 0 of 135°, 90°, 80°, 70°, and 45°, calculated with the
cc-pVTZ basis set. The 1239-1: and RH_F values are in bohr ................................... 95

Table 4. Maximum absolute errors (in eV), relative to MRCI(Q), in the CCSD,
CCSD(T), and CR—CCSD(T) energies for the ground-state PES of the BeFH
system at a Be—F—H angle 0 of 0° (Be inserted between F and H), calculated
with the cc-pVTZ basis set. The R394: and R394; values are in bohr. .................. 96

Table 5. Maximum absolute errors (in eV), relative to MRCI(Q), in the CCSD,
CCSD(T), and CR—CCSD(T) energies for the ground-state PES of the BeFH
system at a Be—F—H angle 6 of 0° (Be approaching the H atom of the HF
molecule), calculated with the cc-pVTZ basis set. The RBe—I-I and RH_F values
are in bohr ................................................................................................................ 97

Table 6. Energies (E) and geometries (R384: and RH_F) of the saddle points
on the BeFH PBS for the Be—F—H angles 0 = 45°, 70°, 80°, 90°, 135°, and
180°, and energies (E) and geometries (R394: and RBe—H) of the HBeF in-
sertion minimum resulting from the CCSD(T), CR—CCSD(T), and MRCI(Q)
calculations with the cc-pVTZ basis set. Energies are in eV, relative to the Be
+ HF asymptote, and internuclear separations are in bohr. .................................... 98

Table 7. Experimental results for the vibrational frequencies of trans-HNOO,
cm‘l, reported in References 104 and 105.. .............................................................. 99

Table 8. Optimized equilibrium geometries of trans-HNOO, resulting from the
force field analysis discussed in Section 5.2. Distances are in A and angles are
in degrees ................................................................................................................ 100

vii

Table 9. Comparison of experimental and theoretical results for the vibrational

frequencies of trans-HNOO, in cm‘l. ................................................................

Table 10. Experimental and theoretical isotopic shifts for the vibrational fre-

quencies of trans-HNOO, in wavenumbers .........................................................

Table 11. Activation energies, AEI, and interallylic distances of the transition

states, R1, for the Cope rearrangement of 1,5-hexadiene ...................................

Table 12. Experimental and theoretical activation and reaction energies, AE1
and AER, respectively, for the Bergman cyclization of System 1, in kcal/mol.

Table 13. Theoretical activation and reaction energies, AEI and AER, respec-

tively, for the Bergman cyclization of System 2, in kcal/mol. ...........................

Table 14. Theoretical activation and reaction energies, AE1 and AER, respec-

tively, for the Bergman cyclization of System 3, in kcal/mol. ...........................

Table 15. Theoretical activation and reaction energies, AE1 and AER, respec-

tively, for the Bergman cyclization of System 4, in kcal/mol. ...........................

Table 16. Theoretical activation and reaction energies, AE1 and AER, respec-

tively, for the Bergman cyclization of System 5, in kcal/mol. ...........................

Table 17. Theoretical activation and reaction energies, AEI and AER, respec-

tively, for the Bergman cyclization of System 6, in kcal/mol. ...........................

Table 18. Theoretical activation and reaction energies, AE1 and AER, respec-

tively, for the Bergman cyclization of System 7, in kcal/mol. ...........................

viii

..... 101

..... 102

..... 103

..... 104

..... 105

..... 106

..... 107

..... 108

..... 109

..... 110

List of Figures

Figure 1. Contour plots of the ground-state PBS for the BeFH system resulting
from the (a) CCSD(T), (b) CR-CCSD(T), and (c) full CI calculations with
the MIDI basis set. The energies E are reported as E + 113.000 hartree. The
thick contour line representing E = —113.930 hartree separates the region
where the contour spacing is 0.010 hartree from the region where the contour
spacing is 0.005 hartree. The thick contour line representing E = —113.800
hartree separates the region where the contour spacing is 0.005 hartree from
the region where the contour spacing is 0.100 hartree. There is a contour
line corresponding to E = —113.931 hartree added to the CCSD(T) PBS to
emphasize the presence of an artificial well, which does not appear on the
CR-CCSD(T) and full CI PESs .............................................................................. 111

Figure 2. The dependence of the differences between the (a) CCSD(T) and
full CI energies, and (b) CR—CCSD(T) and full CI energies for the BeF H
system, as described by the MIDI basis set, on the H—F and Be—F internuclear
separations .............................................................................................................. 112

Figure 3. The potential energy curves of the collinear BeFH system along the
H—F stretch coordinate RH_F at the Be—F distance R3942 fixed at (a) 2.5719,
(b) 5.0, and (c) 8.0 bohr calculated with the (Cl) CCSD(T), (A) CR—CCSD(T),
and (dotted line) full CI approaches, and the MIDI basis set. ............................... 113

Figure 4. The potential energy curves of the collinear BeFH system along the
Be—F stretch coordinate R394 at the H—F distance R34: fixed at (a) 1.7325,
(b) 5.0, and (c) 8.0 bohr calculated with the (Cl) CCSD(T), (A) CR—CCSD(T),
and (dotted line) full CI approaches, and the MIDI basis set. ............................... 114

Figure 5. Contour plots of the ground-state PBS for the BeFH system, at 0 =
180°, resulting from the (a) CCSD(T), (b) CR-CCSD(T), and (c) MRCI(Q)
calculations with the cc—pVTZ basis set. All energies are reported relative to
the Be + HF reactants (R394: 2 50.0 bohr and 123.12 = 1.7325 bohr). The
thick contour line at 1.3 eV separates the region where the contour spacing is
0.3 eV from the region where the contour spacing is 0.5 eV. There is a contour
line corresponding to 0.12 eV added to the MRCI(Q) PBS to emphasize the
depth of the product channel. ................................................................................ 115

Figure 6. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 =
180° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 116

ix

Figure 7. Contour plots of the ground-state PBS for the BeFH system, at 6 =
135°, resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVTZ basis set. All energies are reported relative to
the Be + HF reactants (R384: = 50.0 bohr and RH_p = 1.7325 bohr). The
thick contour line at 1.2 eV separates the region where the contour spacing is
0.3 eV from the region where the contour spacing is 0.5 eV. There is a contour
line corresponding to 0.12 eV added to the MRCI(Q) PBS to emphasize the
depth of the product channel. ................................................................................ 117

Figure 8. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 6 =
135° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 118

Figure 9. Contour plots of the ground-state PBS for the BeFH system, at
6 = 90°, resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVTZ basis set. All energies are reported relative to
the Be + HF reactants (1239-1: 2 50.0 bohr and RH_F = 1.7325 bohr). The
thick contour line at 0.8 eV separates the region where the contour spacing is
0.2 eV from the region where the contour spacing is 0.5 eV. There are extra
contour lines corresponding to 3, 0.23, and 0.09 eV added to the CCSD(T), CR-
CCSD(T), and MRCI(Q) PESs, respectively, to emphasize important features
of the PESs. ........................................................................................................... 119

Figure 10. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 6 =
90° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 120

Figure 11. Contour plots of the ground-state PBS for the BeFH system, at
6 = 80°, resulting from the (a) CCSD(T), (b) CR-CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVTZ basis set. All energies are reported relative to
the Be + HF reactants (R1384: = 50.0 bohr and RH_p = 1.7325 bohr). The
thick contour line at 0.6 eV separates the region where the contour spacing is
0.3 eV from the region where the contour spacing is 0.5 eV. There is a contour
line corresponding to 0.1 eV added to the CR-CCSD(T) PBS to emphasize
the depth of the product channel. .......................................................................... 121

Figure 12. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR-CCSD(T) and MRCI(Q) energies for the 6 =
80° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 122

Figure 13. Contour plots of the ground—state PBS for the BeFH system, at
6 = 70°, resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVTZ basis set. All energies are reported relative to
the Be + HF reactants (R384: = 50.0 bohr and R34: = 1.7325 bohr). The
thick contour line at 0.6 eV separates the region where the contour spacing is
0.3 eV from the region where the contour spacing is 0.5 eV. ................................. 123

Figure 14. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR-CCSD(T) and MRCI(Q) energies for the 6 =
70° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 124

Figure 15. Contour plots of the ground-state PBS for the BeFH system, at
6 = 45°, resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVTZ basis set. All energies are reported relative to
the Be + HF reactants (RBe—F = 50.0 bohr and R34: = 1.7325 bohr). The
thick contour line at 1.3 eV separates the region where the contour spacing is
0.4 eV from the region where the contour spacing is 0.5 eV. ................................. 125

Figure 16. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 6 =
45° BeFH system, as described by the cc-pVTZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 126

Figure 17. Contour plots of the ground-state PBS for the BeFH system, at
6 = 0°, corresponding to Be atom located between H and F, resulting from
the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations with the
cc-pVTZ basis set. All energies are reported relative to the Be + HF reactants
(R384: = 50.0 bohr and RH_F = 1.7325 bohr). A contour spacing of 0.4 eV is
used throughout the plots. ..................................................................................... 127

Figure 18. The dependence of the differences between the (a) CCSD(T)
and MRCI(Q) energies, and (b) CR-CCSD(T) and MRCI(Q) energies for the
6 = 0°, corresponding to Be atom located between H and F, BeF H system,
as described by the cc-pVTZ basis set, on the Be—H and Be—F internuclear
separations .............................................................................................................. 128

Figure 19. Contour plots of the ground-state PBS for the BeFH system, at
6 = 0°, corresponding to H atom located between Be and F, resulting from
the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations with the
cc-pVTZ basis set. All energies are reported relative to the Be + HF reactants
(R384: 2 50.0 bohr and R34: = 1.7325 bohr). A contour spacing of 0.3 eV
is used throughout the plots. The thick contour line corresponding to 3 eV is
added to the PESs to emphasize the presence of an artificially low and well
pronounced barrier on the CCSD(T) PES where none is present ........................... 129

xi

Figure 20. The dependence of the differences between the (a) CCSD(T)
and MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the
6 = 0°, corresponding to H atom located between Be and F, BeFH system,
as described by the cc-pVTZ basis set, on the Be—H and H—F internuclear
separations .............................................................................................................. 130

Figure 21. The dependence of the differences between the MRCI(Q) saddle
point energies and the (solid bars) CCSD(T) saddle point energies, and the
(half-filled bars) CR-CCSD(T) saddle point energies for the BeFH system, on
the angle 6, as described by the cc-pVTZ basis set ................................................ 131

Figure 22. Contour plots of the ground-state PBS for the BeFH system, at 6 2
180°, resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q)
calculations with the cc-pVQZ basis set. All energies are reported relative to
the Be + HF reactants (R394: 2 50.0 bohr and RH_p = 1.7325 bohr). The
thick contour line at 1.3 eV separates the region where the contour spacing is
0.3 eV from the region where the contour spacing is 0.5 eV. There is a contour
line corresponding to 0.099 eV added to the MRCI(Q) PBS to emphasize the
depth of the product channel. ................................................................................ 132

Figure 23. The dependence of the differences between the (a) CCSD(T) and
MRCI(Q) energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 6 2
180° BeFH system, as described by the cc-pVQZ basis set, on the H—F and
Be—F internuclear separations. ............................................................................... 133

Figure 24. The possible mechanisms for the Cope rearrangement of 1,5-
hexadiene ................................................................................................................ 134

Figure 25. The Bergman cyclization reactions. ...................................................... 135

Figure 26. The Cg), cut of the PES of Cope rearrangement of 1,5-hexadiene
as calculated with the UB3LYP, CASSCF, MCQDPT, CCSD(T), and CR-
CCSD(T) methods with the (a) 6-31G* and (b) 6-311G** basis sets. The
energy, AE, is the energy relative to the 1,5-hexadiene reactant molecule.
The solid squares and circles correspond to minima on the CCSD(T) and CR-
CCSD(T) curves, respectively ................................................................................. 136

Figure 27. The CR—CCSD(T) denominator D‘T) (top panel) and the absolute
values of the doubly excited cluster amplitudes t obtained in the CCSD cal-
culations corresponding to the 7bfi —> So: and 5a§ -—> 5b: excitations (bottom
panel), obtained with the 6-31G* basis set, as a function of the interallylic
distance R ............................................................................................................... 137

xii

Figure 28. A comparison of the C2,, cuts of the PES of the Cope rearrangement
of 1,5-hexadiene, as calculated with the MCQDPT, CCSD, CCSD(T), and
CR—CCSD(T) methods, and (a) 6-31G* and (b) 6-311G** basis sets. The
CR—CCSD(T)/D(T) = 1.0 and CR—CCSD(T)/D(T) = 1.4 curves were obtained
by artificially fixing the denominator Dm in the CR—CCSD(T) calculations
at 1.0 and 1,4, respectively. .................................................................................... 138

xiii

1. Introduction

One of the most challenging problems in computational chemistry is the cal-
culation of molecular potential energy surfaces (PESs) which retain high accuracy
throughout all regions of the PES. Especially difficult are regions of the PES where
chemical bonds are broken or formed, and regions which involve diradical structures.
Early ab initio methods developed to obtain energies for atomic and molecular sys-
tems, such as the Hartree-Fock (HF) approach,“4 do not work when bond breaking
or diradical species are involved. The reason behind the failure of HF approaches, in
describing bond breaking or diradical structures, is the neglect of the many-electron
correlation effects in these methods. The HF approaches can obtain most of the elec-
tronic energy (~99%) of the molecular system. However, it is the small percentage of
energy describing the complicated correlated motion of electrons in a molecule which
determines how chemical bonds rearrange in the course of a chemical reaction. It
is for this reason that any method, which is meant to describe reaction pathways,
including diradical and other types of intermediates and transition states, must give
an accurate description of the correlation energy.

The three main approaches currently used to improve upon the HF approaches and
describe the correlation effects in many-electron molecular systems are configuration
interaction (CI),°‘9 many-body perturbation (MBPT) theory,1°‘17 and coupled-cluster
(CC) theorym‘22 The CI method is the simplest way of describing correlation effects.
The CI wave functiOn is a linear combination of the reference (e.g., HF) determinant

and the Slater determinants obtained by exciting electrons from occupied to unoc-

cupied orbitals. If this determinantal expansion is complete (full C1), the correlation
energy and total energy of the molecular system will be exact for the basis set used
in the calculation. Due to the fact that the number of Slater determinants grows fac-
torially with the size of the system, full CI calculations are impractical for systems of
more than a few electrons. For this reason the expansion is usually truncated at some
excitation level. For instance, if the excitations are truncated at only singly and dou-
bly excited determinants we obtain the CISD method. Likewise, if one includes triply
excited determinants along with the singly and doubly excited, and reference determi-
nants then the CISDT method would be obtained and so on. However, the truncated
CI schemes have many problems. One such problem is slow convergence towards the
full CI limit as more highly excited determinants are added, meaning that very long
CI expansions are necessary to obtain acceptable accuracies in calculations. Another
problem with the truncated CI approaches is that they are not size-extensive, i.e. the
truncated CI methods incorrectly describe the dependence of electronic energies on
the size of the system. For the CISD method, the lack of size extensivity can be ap-
proximately accounted for by adding the Davidson corrections.”*24 This procedure is
particularly effective when the quasi-degenerate Davidson corrections”26 are added
to the energies obtained in multi-reference CISD (MRCISD) calculations, although
one must remember that MRCISD calculations are usually prohibitively expensive.
The MBPT approach is probably the easiest way to implement electron corre-
lation into quantum chemical calculations. In MBPT the electronic Hamiltonian is
split into an unperturbed part, which corresponds to the single-determinantal refer-
ence description (usually the HF determinant), and the perturbation which describes

2

the electron correlation. The corrections to the reference wave function and energy
are then calculated from Rayleigh-Schrddinger perturbation theory. If the single de-
terminantal reference wave function for the molecular system is close to the exact
wave function for that system, the convergence of the MBPT series is usually rapid,
making most MBPT calculations quite accurate and relatively inexpensive. Energies
calculated with MBPT are size-extensive, but not variational, and when chemical
bonds are stretched, the MBPT series becomes divergent. Near the equilibrium ge-
ometries of molecules, energies calculated with MBPT are often similar to those of
truncated CI expansions. However, unlike CI, in regions of PESs where bonds are
broken the MBPT energies become unphysical due to the divergent behavior of the
MBPT series.

One of the major drawbacks of MBPT is the necessity to include high-order cor-
rections to obtain a high degree of accuracy in calculations, making the calculations
much more expensive. Coupled-cluster theory can be used to overcome this draw-
back. Due to the exponential form of the CC wave function ansatz, higher-order
excitations are partially included as products of lower-order excitations. Therefore,
higher-order effects can be described at low orders of approximation. In this way, CC
theory can give high accuracy at a relatively low cost. As in the MBPT approach,
the energies calculated with approximate CC methods are non-variational and they
can go below the exact full CI energies. However, the energies obtained with low level
approximations of CC theory are significantly more accurate than the corresponding
CI or MBPT energies, particularly when closed-shell systems are considered. It is also

possible to improve upon the accuracy of low-order CC approximations by combining

3

aspects of MBPT with CC theory. The results of such combined methods will be
shown in later sections of this work.

Coupled-cluster theory has become one of the most popular methods for ac-
curate electronic structure calculations. However, the basic single-reference CCSD
method,”‘29 that is, coupled-cluster theory with singly and doubly excited clusters,
fails in the bond breaking and diradical regions of PESs. The reason for this failure is
the lack of higher-than-doubly excited clusters in the CCSD formalism. Including the
triply, or triply and quadruply excited clusters fully by truncating the CC expansion at
these higher order clusters, the CCSDT”:31 or CCSDTQ methods,32'35 respectively,
is not viable due to the high cost of such calculations. Indeed, the costs of the full
CCSDT and CCSDTQ calculations grow as N 8 and N10, respectively, with the system
size where N is a general measure of molecular size. This prompted the deve10pers
of quantum-chemical methods to design approximate schemes for the inclusion of the
higher-than-doubly excited clusters in CC calculations. One of the most practical,
and widely used, of these approximate schemes has been the CCSD(T)3° approach,
which includes the effect due to triply excited clusters via MBPT (See Refs. 37 and 38
for similar approaches). Another example is the CCSD(TQf)39 method, which adds
a relatively inexpensive MBPT-based correction due to triply and quadruply excited
clusters to the CCSD energy. These CC methods which include the effects of higher-
than-doubly excited clusters have been shown to give the best compromise between
high accuracy and relatively low computer cost in molecular applications.“°*14

Unfortunately, the CCSD(T) and CCSD(TQf) approaches still fail in situations in-

volving bond breaking and diradical molecules.“3*“°‘58 These methods fail because the

4

singly and doubly excited clusters, obtained in the CCSD calculations,which are used
to determine the energy corrections due to triples and quadruples in the CCSD(T)
and CCSD(TQf) approaches, are significantly different from their exact values in re-
gions of PESs where bonds are stretched and diradical structures are present, and
the non-iterative triples and quadruples corrections, which define the CCSD(T) and
CCSD(TQf) methods, fail due to the divergent behavior of the MBPT series at larger
internuclear separations.

At large internuclear separations, when two unpaired electrons are separated by a
large distance, the molecular wavefunction becomes dominated by more than a sin-
gle determinant. For this reason one would normally turn to multi-reference (MR)
variants of the CI, MBPT, and CC theories. However, multi-reference techniques are
not without their own problems, such as a lack of a systematic approach when choos-
ing active orbitals and active electrons (which is usually not a simple task anyway),
higher cost, and, in the case of MRMBPT and MRCC, intruder states. The difficul-
ties in using multi-reference techniques lead to the question of whether it is possible
to describe bond breaking and diradical structures with single-reference CC theory,
which does not require choosing active orbitals in an ad hoc, molecule-by-molecule
manner, and which is free from problems like intruders. The answer is yes, if we use
the method of moments of coupled-cluster equations (MMCC).“9‘°8

In the MMCC theory, a non-iterative energy correction is added to the energy of
a standard CC calculation to provide the exact energy of the system of interest. The
exact MMCC theory, being equivalent to full CI, is too expensive to use, so one has to

use approximate MMCC methods, such as the completely renormalized (CR) CC the-

5

ories,49-55l 57.58.62.64,66—68

These approaches are meant to eliminate the active orbitals
and other elements of multi-reference calculations, and thus retain the “black-box”
character of the standard CC approaches, while also providing high accuracies in bond
breaking and diradical regions of molecular PESs. It has been shown that the CR-
CCSD(T) and CR-CCSD(TQ) methods, where relatively inexpensive non—iterative
corrections due to triply, or triply and quadruply excited clusters, respectively, are
added to the CCSD energy, can provide an excellent description of large fragments of

molecular PESs involving single and double bond dissociation,“9'5°:5‘554‘5662’64’6"”69

and highly-excited vibrational term values near dissociation.“55’6""69

The main goal of this dissertation is to examine the performance of selected CR-
CC methods in calculations of entire PESs for exchange reactions of the type AB
+ C —> A + BC, bond insertion processes, and systems involving diradical species.
A more detailed description of the objectives of this work is given in Section 2. In
Section 3, the original MMCC and CR-CC theories, as formulated in References 49—
51 are discussed in detail. Section 4 shows the results of CR-CCSD(T) calculations
for the PES of the exchange and bond insertion reactions of beryllium with the
hydrogen fluoride molecule, and compares those results to the results of CCSD(T),
MRCI(Q), and full CI calculationsffl’58 Following that, in Section 5, the results of
calculations of the fundamental vibrational frequencies of trans-imine peroxide, using
the CR-CCSD(T) and various other CC approaches, are discussed.70 Trans-imine
peroxide has a partly diradical character and as such it poses a challenge to at least

some electronic structure methods. Finally, in Section 6, the results of applying the

CR-CC approaches to systems involving diradical transition states or intermediates,

6

the Cope rearrangement of 1,5-hexadiene and the Bergman cyclization of enediynes,
are compared to the results of Density Functional Theory (DFT), multi-reference

perturbation theory, CCSD(T), and CCSD(TQf) methods.71

2. Project Objectives

The main objectives of this work are:

A. Applying the CR-CCSD(T) method to the PESs of exchange chemical and
bond insertion reactions, specifically the Be + HF —> BeF + H, BeH + F,
HBeF reactions and comparing the results to standard CC and multi-reference

approaches.57*58

B. Calculating the anharmonic vibration frequencies with CC methods, including
the CR—CCSD(T) approach, of the partly diradical trans-HNOO molecule, in
order to resolve a conflict between two research groups over the assignment of

the experimental vibrational bands.70

C. Applying the CR-CCSD(T) and CR-CCSD(TQ) methods to reactions involving
diradical transition states and intermediates, including the Cope rearrangement
of 1,5-hexadiene and the Bergman cyclization of enediynes, and comparing the
results to those obtained in CASSCF, standard CC, DFT, and multi-reference

perturbation theory calculations.71

3. Theory: Method of Moments of Coupled-Cluster Equations
and the Renormalized Coupled-Cluster Approaches

3.1. The Exact MMCC Formalism

The ground-state wave function IIIIO) of an N-electron system, in the single-

reference CC theory, described by the Hamiltonian H, is defined as

l‘I’ol = eT|<I>), (1)

where |<I>) is the independent particle model reference configuration, usually the
Hartree—Fock determinant, and T is the connected particle-hole excitation Operator,
called the cluster operator. The many-body expansion of the cluster operator T is
typically truncated at a convenient excitation level. The formula defining the cluster

Operator T”) can be written as

mA
TU“ : Z Tn, (2)

11:1
where A denotes the standard single-reference CC approximation (CCSD, CCSDT,
etc.), mA < N is the excitation level characterizing method A, and Tn, n = 1, . . . , m A,
are the many-body components of T”). For example, to obtain the standard CCSD
method, the value of mA is set at 2, to obtain the CCSDT method, mA is set to 3,
and so on.

The cluster operator T”), in the standard CC approximations, is obtained by

9

solving the following system of nonlinear algebraic equations:

Q‘A’H(A’|¢> = 0, (3)

where

H”) = (HeTW)C = e"TWHeTW (4)_

is the similarity transformed Hamiltonian of the approximate CC theory, 62"” is the
projection operator onto the subspace of all excited Slater determinants included in
the cluster operator T“) for the given CC approximation A (e.g., singly and doubly
excited determinants in the CCSD case), and the subscript C designates the connected

part of the corresponding operator expression. The Q“) operator has the form

771/1
Q”) = 2 Q... (5)

n=1
where 62,, is the projection operator onto all n-tuply excited Slater determinants
relative to the reference determinant ((1)) If we insert the CC wavefunction |\Ilo), with
T = T”), into the electronic Schrbdinger equation, and premultiply both sides of the
Schrédinger equation on the left by e‘TW, then project the resulting connected cluster

18, 19,43,49

form of the Schrddinger equation onto the excited configurations included

(A), we obtain the system of CC equations,

in the approximate cluster operator T
Equation (3). This set of equations is then solved for T”) and the CC energy is

finally calculated using

10

E3” = <<I>|H‘A’I<I>>, (6)

which is the connected cluster form of the Schrddinger equation projected onto |<I>).

The exact ground state energy for a given system can be written as

E0 = El,“ + 55;“, (7)

where E0 is the exact, full CI, ground state energy, E5,” is the ground state energy
calculated from the standard CC approximation, Equation (6), and 68A) is the non-
iterative correction which, when added to the energy obtained from the standard CC
approximation, recovers the exact energy. This is the basis of the MMCC formal-
ism; that a correction can be added to the energy of a standard CC approach to
obtain the exact energy of a chemical system. The expression defining 65)") can be
given various forms.”5059*“:°°*°°’72 The form that we use here can be stated as as

f0110WS'49—51’ 55, 62, 66

N n
dimes—El"): Z Z (macs-(mi)MfC(mA>I<I>>/<wole7‘“’l<1>>.(8)

n=mA+l j=mA+l

where

c.-.(m..> = (W6.-.- (9)

is the (n — j)-body component of the CC wave operator 6T“), which defines method

11

A, and

MEC(mA)I<I>> 2Q >-——ZMSC‘”(m )I<I>fi’> (10)

is defined through the coefficients

MSC’(j)(mA) : (¢f{’lH(A)l<I>) (11)

which are projections of the CC equations of the standard approximation A onto the
j-tuply excited Slater determinants IQ?) that are normally not included in method
A. These MSC‘(j)(mA) terms represent the generalized moments of CC equations and
can be easily generated for the basic CC approximations, such as CCSD. If we would
like to recover the exact ground state energy by adding the 63A) to the CCSD energy,
which would be the m A = 2 case, we would calculate the generalized moments of the
CCSD equations corresponding to projections of these equations on triply, quadruply,

pentuply, and hextuply excited determinants, which can be written as

Mé‘ff(2)=<<1>‘-”’£|H°°S°I<I>> (12)
Marc): <<I>rfrleCCSDI<I>> (13)
Militia): <<I> tti‘tIHCCSDIs) (14)
Misfire: (stjbzftilHCCSDtb), (15)

where

12

HCCSD ___ e—(T1+T2)H6T1+T2 : (HeT1+T2)C (16)

is the similarity-transformed Hamiltonian of the CCSD method, léffg), [(1)33321), I‘I’Zbflifi)

and I‘Dadeef ) refer to the triply, quadruply, pentuply, and hextuply excited deter-

ijklmn
minants relative to |<I>), respectively, and 2', j, k,l,m,n, represent occupied spin-
orbitals and a,b, c, d, e, f, represent unoccupied spin-orbitals in |<I>). After calcu-

lating these moments, the M JCC(2)|<I>) quantities can be defined as

M§°(2) =2 Mate) |<I>:3”£>. (17)
i<j<k
a<b<c
MECI2)I<I>>=‘ Z Matte )|<I>§3-”Ef’> (18)
i<j<k<l
a<b<c<d
M§°I2)|<I>)= Z Marge )Isrjbrtz), (19)
i<j<k<l<m
a<b<c<d<e
. abcde abcde
M6CC(2)|(I)> : Z Mijklm{t(2)lq)ijklmjft) ’ (20)
i<j<k<l<m<n
a<b<c<d<e<f

which are used to then calculate the non-iterative correction to the CCSD energy,

N min(n,)6
63C”: 2 Z (‘I’olQnC .. .I2 )MfCI2)I<I>>/I\20IeC+CI<I>). (21)
n=3

Since the Hamiltonian contains only up to two-body interactions, projections of the

CCSD equations onto higher—than—hextuply excited configurations, the Mfc(2)|<I>)

13

quantities with j > 6, vanish and do not need to be considered, even in the exact
case.

The two most important elements of the correction 66A), Equation (8), are the
aforementioned generalized moments and the wavefunction P110). The generalized
moments can be calculated after finding the corresponding cluster operator T”).
However, the form of the wavefunction [\Ilo) in Equation (8) is another issue. If
|\Ilo) is the exact, full CI, ground-state wavefunction, then the full MMCC theory
represented by Equation (8) will recover the exact energy of the system under con-
sideration. In practical calculations, however, this is not possible, since we do not
know the exact wave function (if we knew it, we would not have to do anything).
So one must think of approximate MMCC approaches using simple and easily ob-
tainable forms of I‘llo). The wave functions We) in Equation (21) which define the
R-CCSD(T), CR—CCSD(T), R—CCSD(TQ), and CR—CCSD(TQ) methods and which
enable us to determine the correction 58mm to the CCSD energy are discussed in the

next subsection.

3.2. The Renormalized and Completely Renormalized CCSD(T) and
CCSD(TQ) Methods

As mentioned in the previous subsection, the exact MMCC corrections, 68A) or
683C330, is expressed in terms of the exact wavefunction IIIIO), which is usually not
known. In a general class of approximate MMCC approaches, which are referred to as
the MMCC(mA, m3) schemes, the CI expansion of the wavefunction I‘IIO) is truncated

at the mB-tuply excited Slater determinants relative to the reference |<I>). This means

14

that the MMCC(mA, m3) ground-state energy can be written as” 51:55:56'59"°2'°6

Eg‘MCCmA, mg) 2 E3“ + 60(mA, mg), (22)

with the ground-state correction 60(mA, m3) now having a reduced summation over

n, in which n 3 m3,

60(mA,mB)= Z Z (IIIOIQ,c,,_,-(m,,)MfC(m,,)|¢)/(I110|e7‘""It). (23)

nzmA +1 j:m,1+1

It is clear to see that nonzero values are obtained for the correction 60(mA, m3) only
when my > mA, and the exact MMCC theory is obtained if 7113 = N and III/0) is the
exact full CI wavefunction.

The most practical versions of the MMCC(m A, m B) schemes are those with m A =
2, which are used to correct the results of CCSD calculations. The cost of higher-order
MMCC(2,mB) schemes with 7713 = 5 or 6 is quite high. Fortunately, a great many
chemical systems, including reactions involving the breaking and making of single
bonds and diradicals, can be accurately described using the relatively inexpensive
MMCC(2,3) 0r MMCC(2,4) schemes. We focus only on these two schemes in this

thesis. The energies for the MMCC(2,3) and MMCC(2,4) schemes can be written as

EIIMCCQJ) = EECSD + (‘PolQa M§C(2)l@>/(‘I’oleT‘+T2|¢)I (24)
and

15

ESJMCCQA) = EEC” + (‘Po|{Q3 M§C(2) + Q4 [Mfc(2) + T1M3CCI2)I}|<I>>

/ (\I’oleT‘+T" l¢> , (25)

respectively, where E80513 is the CCSD energy, and the Macc(2)|<1>) and Mfc(2)|<I>)
quantities were defined in the previous subsection (cf. Equations (17) and (18),
respectively).

The R—CCSD(T) and CR-CCSD(T) methods are built from the MMCC(2,3)
scheme, whereas the R—CCSD(TQ) and CR-CCSD(TQ) approaches are built from
the MMCC(2,4) scheme. These R- and CR-CC methods are obtained when a low-
order MBPT expression is used to define the ground-state wavefunction l‘IIO) in the
MMCC(mA, m3) energy formula. The energy formula for the CR—CCSD(T) method

is49—52, 54, 55, 57, 62, 66

EOCR-CCSMT) : Egcsp + (\IICCSD(T)|Q3 Mfcf2llq’) / (QCCSD(T)|8T1+T2I(I)>, (26)

(IICCSD

where the | (T)) wavefunction is the simple MBPT—like expression

IIpCCSD<T)) = (1 + T1 + T2 + R33)(VNT2)C + RgC’VNT,)|<I>), (27)

with T1 and T2 being the cluster operators obtained by solving the CCSD equations,

R33) represents the three—body component of the MBPT reduced resolvent, and VN

16

designates the two-body part of the Hamiltonian in the normal product form, H N =

H - (<I>|H|<I>). If we replace the M°°C(2) moments in the CR-CCSD(T) method with

z'jk
only their lower-order estimates, we obtain the R-CCSD(T) energy formula, which is

written as follows“9“52~545533745166

Est-CCSD(T) : EECSD + (\IJCCSD(T)|Q3 (VNT2)C|(I))/(‘IJCCSD(T)|6T1+T2I‘D) ' (28)

We can rewrite the R— and CR-CCSD(T) energies in the following more compact

forms,

Bil-CCSD(T) = EOCCSD + N(T)/D(T), (29)

EEK-CCSD(T) : Egcso + NCR(T)/D(T), (30)

where the numerators N (T) and N CR”) are defined as

Nm = <¢l<TJVN)CR(()3)(VNT2)Cl¢>+<©l(T1tVN)CRt()3)(VNT2)Clq)> (31)

NCRC" = <<I>I(T2*V~)CRS3’M§°(2)I<I>>+<<I>|(T1*V~)c ‘3’M§°(2)I¢>, I32)

and the denominator D”) is defined as

17

D(T) : <\PCCSD(T)|6T1+T2|(I)>. (33)

When the energy equations are written in the above form, the relationship between
the CR—, R—, and standard CCSD(T) methods is easy to see. If we replace the
denominator Dm in the R—CCSD(T) energy formula with 1, we obtain the standard
CCSD(T) energy formula. This is a justifiable step from the MBPT point of view,

since if one looks at the explicit form of the denominator Dm,

0““) = 1+ (<1>|T,"T,|<I>) + (<I>|T,i (T2 + §T12) |<I>)
+I<I>IITJVN)CRIC’IT1T2 + IT?)

+ITIVN)CRIC’ITIT2 + %T3)|<I>>. (34)

it is simply equal to 1 plus terms which are at least second order in the perturbation
VN. It is this denominator Dm, which “renormalizes” the triples correction N (T)
or N (mm to the CCSD energy, that is responsible for overcoming the failure of the
standard CCSD(T) method in cases of bond—breaking and diradical systems.

The idea of renormalizing the CCSD(T) approach can also be extended to the
CCSD(TQ) method, in which the CCSD energy is corrected for the effect of triply
and quadruply excited clusters. The renormalized CCSD(TQ) method is based on
the more general MMCC(2,4) approximation scheme. There are two particularly use-

ful variants of the CR—CCSD(TQ) approach, which are labelled “a” and “b”. These

18

CR—CCSD(TQ),a and CR-CCSD(TQ),b energies are calculated by the following for-

mulas49—52, 54, 55, 62, 66

BER-CCSD(TQLX : EECSD + <wCCSD(TQ),x|Q3 M3CC(2) + Q4[T1M§:C(2) + M4CC(2)H(I)>/

(\pCCSD(TQ),x|eT1+T2lq)> (x : a, b), (35)
where
lqlccsmTanI) = l‘I’CCSD(T)> + %T2T2(1)I<I>), (36)
and
IIIICCCWC“) = I2CCCD‘T’) + %T§I<I>>, (37)

with T5” representing the first-order MBPT estimate of T2 and IIIICCSmTl) defined
by Equation (27). The energy formulas for the CR—CCSD(TQ),x approaches can be

rewritten, much like the CR-CCSD(T) formulas, in the following way

EgR-CCSDWQLX = Egcso + NCR(TQ),X/D(TQ),x (x = a, b), (38)

where

NCR‘CC’IC = NCRC’ + %<<I>ITJ(Té”)*lT1M§°(2) + MfCI2)II<I>>, (39)

19

NCC‘CC"C -—- NCC‘C’ +(%I<I)| TJ)IT1M§CI2) + MfCI2)II<I>>. (40)
DITW = D“ +—;(<1>|T,i(T._§1 ) (17113497127:2 +— 21—4T1“)|<I>), (41)
D‘CC’I" = D‘C+ %<<1>|(TJ)I%T22+ ITETz + ;—.T;‘)I<I>>. (42)

with N CR”) and DIT) are defined by Equations (32) and (34), respectively. One can
also simplify the CR-CSD(TQ),x energy expressions by considering only the lowest or-
der estimates of the ij°f(2 ) 2and ij°[fl(2 ) moments, which enter the M300(2)|<I>) and
Mfc(2)|<1>) quantities, and dropping the T1M3(2) term in Equation (35), which gives
us the R—CCSD(TQ),x (x=a,b) variants of the MMCC theory. As in the R—CCSD(T)
method, one can obtain the usual CCSD(TQ) methods, such as CCSD(TQf), from the
R—CCSD(TQ) approaches by replacing the denominator D(TQ).x in the R-CCSD(TQ)
energy expressions with 1 and replacing the numerator N CR(TQ)”‘ with one of its low
order estimates.

The relatively simple relationships between the standard CCSD(T) and CCSD(TQ)
approaches and their CR-CCSD(T) and CR—CCSD(TQ) counterparts imply that the
computer costs between the standard and CR approaches are nearly identical. The
cost of the standard CCSD(T) method is ngnfi (where no is the number of occupied

orbitals in the basis and nu is the number of unoccupied orbitals in the basis) in the

iterative CCSD steps and 722an in the non-iterative steps for the triples correction.

20

The CR—CCSD(T) approach also scales as 11377.: in the iterative CCSD steps and 713723
in the non-iterative triples correction. Precisely, the cost of the non-iterative triples
correction for the CR-CCSD(T) method is twice as large as that of the standard
CCSD(T) approach. Likewise, both the standard and CR-CCSD(TQ) approaches
4

scale as ngn: in the iterative CCSD steps, ngn in the non-iterative triples correc-

tions, and ngnf’, in the non-iterative quadruples corections, with the non-iterative
triples and quadruples corrections for the CR-CCSD(TQ) approaches being twice as
costly as the corrections for the standard approach. This means that by using the
CR—CC methods, one can gain significantly higher accuracy of results, especially in
the bond breaking and diradical regions of PESs, over the standard CC approaches

without giving up very much in computer costs, as will be shown in the following

sections.

21

4. Applications: Molecular Potential Energy Surfaces for the
Be + HF Reaction

4.1. Background Information and Motivation

The Be + HF reaction has been the subject of several theoretical studies, includ-
ing calculations of the ground-state PES for the BeFH system with density func-

74 ~76

tional theory,73 diatomics—in-molecules, and the configuration interaction (CI)

approach.73'77'7‘3 The PES for the BeF H system has been fit to different functional

77-81 and the reaction dynamics have been calculated.77'8° The BeFH system

forms
provides an excellent test case for new electronic structure methods aimed at produc-
ing PESs which can accurately describe the making and breaking of chemical bonds,
since the BeFH system is the lightest member of an important family of exchange
reactions involving alkaline earth metals and halides.82”93 It has been shown previ-
ously that the renormalized CC approaches are capable of removing the failings of the

standard CC approaches for unimolecular dissociations,49“”2'54:55:"’£”‘32""4"3‘3’69 diradi-

94’9° and highly excited vibrational states near the dissociation threshold.54’55~°°'69

cals,
However, it is also important to show that these renormalized methods are also ca-
pable of obtaining accurate PESs for exchange reactions of the A + BC —> AB + C
type, including the reactant channel, the product channels, and the region of non-
interacting atoms. As shown below, the Be + HF reaction also has a bond-insertion

channel that leads to the formation of HBeF, along with the more usual BeF + H

and BeH + F channels.

22

4.2. Computational Details

The BeFH system was studied using three different basis sets, the MIDI,96 cc-
pVTZ, and cc-pVQZ basis sets.97 In the case of the small MIDI basis set, using a
total of only 14 basis functions, the results of the standard CCSD, standard CCSD(T),
and the renormalized CCSD(T) methods (both the R-CCSD(T) and CR-CCSD(T)
variants) were compared against the results of the exact calculations using the full CI
method. In the case of the larger cc-pVTZ and cc-pVQZ basis sets, the R-CCSD(T)
and CR-CCSD(T) results are compared with the results of the Q corrected MRCI
calculations, as described below. In all calculations involving the MIDI basis set, the
lowest 10 orbital (~13 orbital of F) was kept frozen.

The large cost of the full CI calculations made calculating the entire 3-dimensional
ground-state PES of the BeFH system very expensive, even with the small MIDI basis
set, so the Be—F-H angle, 6, was fixed at zero degrees, meaning that only the collinear
approach of the Be atom to the F atom of the HF molecule from the fluorine side was
considered in the case of the MIDI basis set. In this case, the electronic energies were
calculated on a grid of 345 points consisting of 23 Be—F distances R394, namely,
R39_F=1.8, 1.9, 2.0, 2.2, 2.4, 2.5, 2.5719, 2.6, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.5,
4.7, 5.0, 5.2, 5.5, 6.0, and 8.0 bohr and 15 H-F distances R342, namely, RH_F=1.2,
1.4, 1.6, 1.7325, 1.8, 2.0, 2.25, 2.5, 2.75, 3.0, 3.5, 4.0, 5.0, 6.0, and 8.0 bohr. The
CCSD, CCSD(T), R-CCSD(T), CR-CCSD(T), and full CI energies were calculated
at each nuclear geometry in this basic grid. Other points were considered only if

more detailed information about the PES was needed, for example in the saddle

23

point region.

For calculations involving the larger cc-pVTZ basis set, consisting of 85 basis func-
tions, the CCSD, CCSD(T), R—CCSD(T), CR-CCSD(T), and MRCI calculations were
performed. The CC calculations were performed using the CC codes°9 which have
been implemented by the Piecuch group in the GAMESS package.98 The correspond-
ing MRCI calculations were carried out using the internally contracted MRCI(Q)
approach, developed by Werner and Knowlesgg'100 and implemented in MOLPRO101
which includes quasi-degenerate Davidson corrections, and employs a complete active
space self-consistent field (CASSCF) reference. The CASSCF active space used as a
reference for these MRCI(Q) calculations consisted of 8 active orbitals and 8 active
electrons, corresponding to the 2s and 2p shells of beryllium, the 2p shell of fluorine,
and the Is shell of hydrogen.

In the case of the cc-pVTZ basis set, the entire 3-dimensional ground-state PES
of the BeFH system was calculated, probing three different product channels, BeF
+ H, BeH + F, and the formation of HBeF. The PES was calculated on a grid of
2852 nuclear geometries which were defined as follows. We used seven values of the
Be—F—H angle 6 ranging from 0° to 180° (6: 0°, 45°, 70°, 80°, 90°, 135°, and 180°;
where 6=180° corresponds to a collinear arangement of the Be, F, and H atoms with
the F atom between the Be and H atoms). For values of 6 ranging from 6=45° to
6=180°, the following Be—F and H—F distances, R394: and 1211.12, respectively, were
used: R3942: 1.8, 1.9, 2.0, 2.2, 2.4, 2.5, 2.5719, 2.6, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9,
4.1, 4.5, 4.7, 5.0, 5.2, 5.5, 6.0, and 8.0 bohr, and 123.1»: 1.2, 1.4, 1.6, 1.7325, 1.8,
2.0, 2.25, 2.5, 2.75, 3.0, 3.5, 4.0, 5.0, 6.0, and 8.0 bohr. For 6=0°, two different

24

arrangements of the Be, F, and H atoms were considered. The first arrangement for
6=0° was the beryllium atom approaching the hydrogen atom of the HF molecule;
in this case, the Be—H distances R394; considered were R3941: 1.8, 1.9, 2.0, 2.2,
2.4, 2.5, 2.52, 2.6, 2.7, 2.9, 3.1, 3.3. 3.5, 3.7, 3.9, 4.1, 4.5, 4.7, 5.0, 5.2, 5.5, 6.0, and
8.0 bohr, and the H—F distances considered were RH_F= 1.2, 1.4, 1.6, 1.7325, 1.8,
2.0, 2.25, 2.5, 2.75, 3.0, 3.5, and 4.0 bohr. The other arrangement of atoms for 6=0°
had the beryllium atom inserted between the fluorine and hydrogen atoms; for this
arrangement the Be—F and Be—H distances considered were RBe_p= 1.8, 1.9, 2.0, 2.2,
2.4, 2.5, 2.5719, 2.6, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.5, 4.7, 5.0, 5.2, 5.5, and
6.0 bohr, and R3941: 1.8, 1.9, 2.0, 2.2, 2.4, 2.5, 2.52, 2.6, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7,
3.9, 4.1, 4.5, 4.7, 5.0, 5.2, 5.5, 6.0, and 8.0 bohr. Once again, additional points were
considered in PES regions of special importance, such as the saddle point region.

In the case of the largest cc-pVQZ basis set, consisting of 175 orbitals, the CCSD,
CCSD(T), R-CCSD(T), CR-CCSD(T), and MRCI(Q) calculations were performed
only for the collinear arrangement of the Be, F, and H atoms corresponding to 6=180°.
The PES calculated with the cc-pVQZ basis set used the following Be—F and H—F
distances: 12394:: 1.8, 1.9, 2.0, 2.2, 2.4, 2.5, 2.5719, 2.6, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7,
3.9, 4.1, 4.5, 4.7, 5.0, 5.2, 5.5, 6.0, and 8.0 bohr, and R342: 1.2, 1.4, 1.6, 1.7325, 1.8,

2.0, 2.25, 2.5, 2.75, 3.0, 3.5, 4.0, 5.0, 6.0, and 8.0 bohr.

4.3. Results of the Standard, Renormalized, and Completely
Renormalized CCSD(T) Calculations Employing the MIDI Basis Set

The results of the calculations of the ground-state PES of the BeFH system can be

25

seen in Figures 1—4. The CCSD, CCSD(T), R-CCSD(T), CR-CCSD(T), and full CI
energies, calculated at each of the 345 nuclear geometries can be found in Appendix
A. The maximum absolute errors, relative to full CI, for the CCSD, CCSD(T), R-
CCSD(T), and CR—CCSD(T) results can be seen in Table 1. It can be seen from Table
1, as well as Appendix A, that the CCSD energies have large errors, with respect to the
full CI energies, over the entire potential energy surface. The R-CCSD(T) approach
improves upon the CCSD(T) results, but does not give results which are as high in
quality as the CR—CCSD(T) method, as can be seen in Table 1 and Appendix A.
For these reasons this discussion will focus on comparing the standard CCSD(T) and
CR—CCSD(T) energies, with those of the full CI calculations.

By comparing the contour plots in Figures 1(a) and 1(c), it is easy to see that the
PES calculated with the CCSD(T) method, Figure 1(a), clearly differs from the full
CI PES, Figure 1(c). The differences between the two contour plots are particularly
large when both the Be——F and H—F bonds are stretched. These differences are greater
than 10 millihartree for a large region of the surface, the entire region of R3e_p 2
3.9 bohr and RH_F 2 6.0 bohr, and the RBe_p 2 3.3 bohr and R34: 2 8.0 bohr
region. The errors for the CCSD(T) PES are greater than 5 millihartree in the entire
R394: > 3.0 bohr and RH_F 2 5.0 bohr region, and the region of R394: = 1.8—2.0
bohr and R34: 2 275—30 bohr. This should be contrasted with the fact that there
are no differences between the CR-CCSD(T) and full CI data which are greater than
, 3.1 millihartree. In fact, there are only four nuclear geometries in the entire grid
of points used by us where the error associated with the CR—CCSD(T) method is

between 3.0 and 3.1 millihartree. Small errors of 2—3 millihartree are observed for

26

the CR—CCSD(T) PES only in the region of R394: 2 3.3 bohr and RH_F 2 4.0 bohr.
For most of the remaining geometries, the errors in the CR—CCSD(T) results are
approximately 1 millihartree or less. For this reason, the contour plot resulting from
the CR-CCSD(T) calculations, Figure 1(b), is almost identical to the contour plot
resulting from the full CI calculations, Figure 1(c).

The dependence of the differences between the CCSD(T) and the full CI results on
the nuclear geometry is shown in Figure 2(a), and the dependence of the differences
between the CR—CCSD(T) and the full CI results on the nuclear geometry is shown
in Figure 2(b). It can be seen in Figure 2(b) that the CR—CCSD(T) PES is located
approximately 1—2 millihartree above the full CI PES, and the surfaces are nearly
parallel to each other. This should be contrasted with the large and highly nonuniform
distribution of errors seen from the CCSD(T) results shown in Figure 2(a). One should
also notice that most of the CCSD(T) surface is located below the full CI surface.
The PES obtained from the CCSD method lies entirely above the full CI surface, but
it is far from being parallel to the full CI PES, and the errors associated with the
CCSD PES are quite large. In fact, the entire region corresponding to R384: 2 3.3
bohr and R34: 2 5.0 bohr on the CCSD PES lies over 10 millihartree above the full
CI PBS, with a maximum error of 16.3 millihartree at R3e_p = 3.9 bohr and RH_F =
8.0 bohr. The maximum absolute error in the CCSD(T) PES, of 28.6 millihartree, is
even greater than that of the CCSD PES. The CR-CCSD(T) PES shows a remarkable
reduction in these maximum errors, to a mere 3.1 millihartree. A comparison of these
errors can be seen in Table 1. The R—CCSD(T) approach shows relatively small

errors when compared to the errors from the CCSD and CCSD(T) methods over the

27

entire PES, and even compared to the CR-CCSD(T) approach when the internuclear
distance of one of the diatomics, BeF or HF, is close to the corresponding equilibrium
bond length. When the Be—F and H~F bonds become stretched, however, the errors
associated with the R-CCSD(T) surface grow larger than those associated with the
CR—CCSD(T) surface, even though they do stay well below the errors inherent in
the CCSD and CCSD(T) surfaces. This increase in errors in the R-CCSD(T) PES
makes it less parallel to the full CI surface, compared to the surface calculated with
the CR—CCSD(T) method.

It is interesting to notice that stretching the H—F bond has a more drastic effect
on the results of the CCSD and CCSD(T) calculations than stretching the Be—F
bond. For example, if the H—F bond is stretched to 3.0—5.0 bohr, and all Be—F
distances are considered, the errors in the CCSD and CCSD(T) results are 13.163
and 9.690 millihartree, respectively. When the Be—F bond is stretched to 3.1—5.0
bohr and all H—F distances are considered, the errors in the CCSD and CCSD(T)
results are much larger, 16.287 and 27.887 millihartree, respectively. This is not
the case for the CR-CCSD(T) method; the errors for the CR-CCSD(T) approach
remain small, independent of the region of the PES under consideration. In fact, the
mean absolute error over the entire PBS for the CR—CCSD(T) method is only 0.9
millihartree, while the mean absolute errors for the CCSD and CCSD(T) PESS are
4.0 and 2.5 millihartree, respectively.

The CCSD(T) PES has the wrong asymptotic behavior in the BeF + H and Be
+ F + H regions. This can be seen by comparing Figures 1(a) and 1(c), and by

analyzing Figure 3. In Figure 3, the CCSD(T), CR-CCSD(T) and full CI potential

28

energy curves have been plotted as a function of the H—F bond distance, for three
representative values of the Be—F distance. The fixed Be—F distances chosen were:
the equilibrium bond length of BeF, R394: 2 2.5719 bohr, corresponding to the BeF
+ H product channel, R394: 2 5.0 bohr, and R394: 2 8.0 bohr. For the BeF distance
fixed at 2.5719 bohr, there is a significant increase in the errors characterizing the
CCSD(T) curve; see Figure 3(a). The small < 1 millihartree error corresponding to
H—F distances less than 3.0 bohr, increases to 4 millihartree for distances greater than
5.0 bohr. The CCSD(T) curve in Figure 3(a) shows a well-pronounced, unphysically
large hump, which is much smaller in the full CI curve. The CR-CCSD(T) curve
shown in Figure 3(a) has an identical shape to the full CI curve, and the errors
associated with the CR—CCSD(T) method are less than or equal to 1 millihartree.
For the case of R394: 2 5.0 bohr, shown in Figure 3(b), once again, the small errors
in the CCSD(T) results, less than 0.5 millihartree for distances less than 3.5 bohr,
become large as the H—F bond distance increases. They increase to 9.4 millihartree at
an H—F distance of 5.0 bohr and to 27.9 millihartree at an H—F distance of 8.0 bohr.
Also, the CCSD(T) curve goes well below the exact, full CI curve as the H—F bond
distance becomes large. On the other hand, the CR—CCSD(T) results for R384: =
5.0 bohr show very small errors when compared to the full CI results, in the range of
0.2 to 2.6 millihartree, and the CR-CCSD(T) curve, as in the case of R384 2 2.5719
bohr, has virtually the exact shape of the full CI curve. For the case of R3e_p =
8.0 bohr, shown in Figure 3(c), the CR-CCSD(T) and full CI results agree to within
0.3—1.8 millihartree for the entire curve, while the CCSD(T) results Show large errors
at large H—F distances.

29

In Figure 4 the CCSD(T), CR-CCSD(T), and full CI potential energy curves are
plotted for a few fixed values of the H—F bond distance, as a function of the Be—F
bond distance. The fixed values of the H—F bond distance are R34: = 1.7325 bohr,
which is the equilibrium bond length in the HF molecule, R34 = 5.0 bohr, and R34:
= 8.0 bohr. When the H—F bond distance is fixed at its equilibrium bond length,
the CCSD(T), CR-CCSD(T), and full CI curves are all virtually identical, as can be
seen in Figure 4(a). This is a consequence of the closed shell nature of the BeFH
system in this reactant region of the PES. However, when the H—F bond distance is
stretched to 5.0 or 8.0 bohr, the CCSD(T) results become drastically worse, as shown
in Figures 4(b) and 4(c). For the R34: value of 5.0 bohr (see Figure 4(b), the errors,
relative to full CI, in the CCSD(T) method are 3.5 millihartree at R384: = 2.5719,
9.7 millihartree at R394; 2 5.5 bohr, and 9.1 millihartree at R394: 2 8.0 bohr. For
the case where the R34: distance is fixed at 8.0 bohr, the errors for the CCSD(T)
approach become 4.3 millihartree at R394: = 2.5719, 28.6 millihartree at R394: = 5.5
bohr, and 21.7 millihartree at R394: 2 8.0 bohr, which can be seen in Figure 4(c). In
both cases, when the H—F distance is fixed at R34: 2 5.0 or 8.0 bohr, the CCSD(T)
curves are located below the full CI curves. Meanwhile, the CR—CCSD(T) method
eliminates the failings of the CCSD(T) approach for both the R34: value fixed at 5.0
bohr and at 8.0 bohr. In both cases, the CR—CCSD(T) curves are virtually identical
to the full CI curves, as can be seen in Figures 4(b) and 4(c). When the H—F distance
is fixed at 5.0 bohr, the errors for the CR-CCSD(T) approach are 0.7 millihartree at
R384: 2 2.5719 bohr, 2.3 millihartree at R394: = 5.5 bohr, and 1.8 millihartree at

R384: = 8.0 bohr. The errors for the H—F distance fixed at 8.0 bohr do not change

30

significantly from the values obtained in the R34: = 5.0 bohr case.

The CCSD(T) PES shows an artificial maximum at R394: z 4.5 bohr and R34:
2 5.0 bohr, as can be seen by inspecting the contour line of -113.890 hartree in Figure
1(a). This hump creates the impression that there is a barrier to the formation of
the strange BeF product with a stretched Be—F bond. Clearly, this is an unphysical
behavior which is not consistent with the full CI PES, shown in Figure 1(c). By
comparing the thick contour lines corresponding to -113.930 hartree for the CCSD(T)
PES, Figure 1(a), and the full CI PES, Figure 1(c), one can clearly see that the
CCSD(T) method allows for product formation at a lower energy than the exact, full
CI method. The product valley of the CR-CCSD(T) PES is shaped nearly identically
to the product valley of the full CI surface, as can be seen by inspecting the contour
lines at —113.930 and -113.925 hartree in Figures 1(b) and 1(c). The CR—CCSD(T)
approach is also able to eliminate the unphysical barrier at R394: z 4.5 bohr and
123.1: x 5.0 bohr, which appears on the CCSD(T) surface but not on the full CI
surface. In addition to the failures of the CCSD(T) approach that were mentioned
above, the CCSD(T) method also produces a pronounced saddle point at R384: z
2.8 bohr and RH_F z 3.5 bohr. This saddle point could be found on neither the
exact, full CI surface nor on the CR-CCSD(T) surface, despite the calculation of
many additional nuclear geometries in that region. The most likely reason for the

absence of the saddle point in the full CI PES is the small size of the basis set used.

31

4.4. Results of The Standard, Renormalized, and Completely
Renormalized CCSD(T) Calculations Employing the cc-pVTZ and
cc—pVQZ Basis Sets

As mentioned above, the CC and MRCI(Q) PESs calculated with the cc-pVTZ
basis set were obtained for several values of the Be—F—H angles, 6, whereas the CC
and MRCI(Q) calculations for the largest cc-pVQZ basis set were limited to 6 =
180°. Our discussion focuses on the performance of the CCSD and CCSD(T) vs
MRCI(Q) and the CR—CCSD(T) vs MRCI(Q) methods. We begin the discussion
with the calculations for 6 2 180° and the cc-pVTZ basis set.

The CCSD(T), CR-CCSD(T), and MRCI(Q) ground-state PESs for the BeF H
system at 6 = 180°, calculated with the cc-pVTZ basis set, are shown in Figure
5. The maximum absolute errors, relative to the MRCI(Q) energies, for the CCSD,
CCSD(T), and CR-CCSD(T) methods can be found in Table 2. As can be seen from
Table 2, the CCSD PES differs greatly from the PES calculated with the MRCI(Q)
approach. The errors are especially large in the regions where the Be—F and H—F
bonds are stretched. The differences between the CCSD and MRCI(Q) energies are
greater than 0.5 eV in the entire region of R384: 2 3.9 bohr and RH_F 2 4.0 bohr,
and for almost all geometries in the RBe_p < 3.9 bohr and 2.5 bohr S 1211.}: < 4.0
bohr region.

Comparing the results of the CCSD(T) and MRCI(Q) calculations, Figure 5 and
Table 2, one can see that, much like the CCSD PES, the CCSD(T) PES differs greatly
from the MRCI(Q) PES. Once again the errors are quite large when the Be—F and H—

F bonds are stretched. The errors, relative to the MRCI(Q) results, for the CCSD(T)

32

energies are greater than 1 eV for the entire R384: 2 5.5 bohr and R54: 2 6.0 bohr
region, and they are greater than 0.5 eV for the entire R394: 2 5.0 bohr and R34:
2 5.0 bohr region. The errors are greater than 0.2 eV for most nuclear geometries
in the R394: < 2.5 bohr and RH_F 2 2.5 bohr , the R394: 2 3.5 bohr and R114: 2
3.5 bohr, and the 2.5 bohr g RBe_p < 3.5 bohr and RH_F z 3.0 bohr regions. The
maximum differences, for the entire 6 2 180° PES, between the CCSD and MRCI(Q),
and CCSD(T) and MRCI(Q) results are 1.137 and 3.269 eV, respectively.

The poor performance of the CCSD and CCSD(T) approaches should be con-
trasted with the excellent performance of the CR-CCSD(T) approach. The differ-
ences between the CR-CCSD(T) and MRCI(Q) energies are typically in the range of
001—01 eV for the collinear, 6 2 180°, BeFH system, as described by the cc-pVTZ
basis set. In fact, the maximum absolute error for the entire CR—CCSD(T) PES is
very small (0.180 eV). The CR-CCSD(T) surface is located slightly above the PES
calculated with the MRCI(Q) approach, in direct contrast with the results of the
CCSD(T) calculations, which produce energies way below the MRCI(Q) energies, as
can be seen by inspecting Figures 5 and 6. There is a large nonuniform distribution
of errors in the CCSD(T) surface, which can be seen in Figure 6(a), while the errors
for the CR—CCSD(T) surface remain consistently small over all nuclear geometries,
as seen in Figure 6(b).

The CCSD(T) method produces an artificial barrier which leads to the Be + F +
H atomic products. This barrier does not appear on the MRCI(Q) and CPI-CCSD(T)
PESs, as can be seen in Figure 5. The shallow van der Waals well, located in the

product valley, lies below the energy of the reactants in the case of the CCSD(T)

33

approach, while for the MRCI(Q) and CR-CCSD(T) methods the shallow van der
Waals well in the product channel lies above the energy of the reactants. Again,
this can be seen by inspecting Figure 5. In the case of CR-CCSD(T), the product
valley, corresponding to the BeF + H products, and the region of separated atoms,
corresponding to the Be + F + H channel, are shaped in the same way as in the
MRCI(Q) PES, as can be seen by examining the contour lines corresponding to 1.3
eV, 5.3 eV, and 5.8 eV in Figures 5(a) and 5(c). Unlike the CR-CCSD(T) and
MRCI(Q) surfaces, the energy value of 5.8 eV is never reached on the CCSD(T) PES;
the maximum energy on the CCSD(T) surface is 5.4 eV. The endothermicity of the Be
+ HF -—> BeF + H reaction, as computed with the CCSD(T) results is -0.009 eV. This
value has an incorrect sign when compared to the value calculated with MRCI(Q)
of 0.140 eV and the MRCI results reported by Aguado et al.73 of 0.26 eV, as well
as the experimentally derived value of 0.193 eV from the binding energies of BeF102
and HF.”3 The endothermicity value of 0.284 eV calculated from the CR-CCSD(T)
results, retains the correct sign and it is in good agreement with the value reported
in Reference 73, although it is slightly higher than our calculated MRCI(Q) value.
The results for the ground-state PESs obtained in the CCSD, CCSD(T), CR-
CCSD(T), and MRCI(Q) calculations, employing the cc-pVTZ basis set, for the Be—
F—H angles 6 of 45°, 70°, 80°, 90°, and 135°, can be found in Figures 7—16 and Table 3.
The CCSD and CCSD(T) surfaces for each of these angles show large errors relative
to the MRCI(Q) surface, as can se seen in Figures 7—16, especially at larger Be-F
and H——F separations. The maximum error, relative to the MRCI(Q) approach, in the

CCSD results, for the angles 45°—135°, is 1.284 eV. The error in the CCSD results

34

seems to be almost invariant with respect to the angle 6, as can be seen in Tables 2
and 3. This is not the case, however, when the CCSD(T) results are examined. In
the case of the CCSD(T) method, as the angle 6 decreases, the maximum absolute
error increases drastically, to a whopping 10.988 eV when R384: 2 8.0 bohr, R34: 2
8.0 bohr, and 6 = 45°, as shown in Table 3 and Figure 16(a). For Be—F—H geometries
which are significantly bent, 6 S 90°, there is a larger effect on the results of the
standard CCSD(T) method when the Be—~F bond is stretched than when the H—F
bond is stretched. An example of this behavior can be seen when 6 = 70°. When the
Be—F bond is stretched to 3.1—5.0 bohr and all H—F bond distances are considered,
the maximum absolute error in the CCSD(T) results is 0.294 eV, while when the H—F
bond is stretched to 3.0—5.0 bohr and all Be—F bond distances are considered, the
maximum absolute error in the CCSD(T) results, relative to MRCI(Q), is 2.581 eV,
as can be seen in Table 3. These failures in the CCSD and CCSD(T) results contrast
with the exceptional results from the CR—CCSD(T) calculations. The maximum
absolute error for the CR-CCSD(T) results, relative to MRCI(Q), for the angles 6
= 45°—135° is 0.407 eV, corresponding to 6 = 45°. The errors in the CR-CCSD(T)
results do not vary significantly as the angle 6 decreases, unlike the errors in the
standard CCSD(T) calculations. In fact, if we limit the Be—F—H angle to the range
of 6 = 70°—180°, the maximum absolute errors in the CR—CCSD(T) results fluctuate
around 0.2 eV and seem virtually indepedent of 6, as shown in Tables 2 and 3. It is
also important to note that the CR-CCSD(T) PES has a quasi-variational character:
it lies slightly above the MRCI(Q) PES, while the CCSD(T) surface lies mostly below
the MRCI(Q) PES (see Figures 6, 8, 10, 12, 14, 16).

35

The PES for the BeF H system at 6 = 90° provides a good example of how the
CPI-CCSD(T) method can provide results that are of MRCI(Q) quality, when the
CCSD(T) method gives results that are qualitatively incorrect (see Figure 9 and
Table 3). At larger Be—F and H—F distances, the CCSD(T) results for 6 = 90° show
large errors relative to the MRCI(Q) results. These errors become as large as 4.142
eV, as shown in Table 3. The CCSD(T) method also produces an artificial well ~2.5
eV deep near the nuclear coordinates 1239-1: = 6.0 bohr and RH_F = 6.0 bohr, which
can be seen by inspecting the contour line at 3 eV in the CCSD(T) PES shown in
Figure 9(a). This well does not appear on either the CR-CCSD(T) or the MRCI(Q)
PESs. The CR-CCSD(T) and MRCI(Q) PESs do, however, show the presence of a
well in the product channel, as can be seen by examining the contours labelled 0.23
and 0.09 eV in Figures 9(b) and 9(c), respectively. This well is due to the beginning
of the insertion of the beryllium atom between the hydrogen and fluorine atoms, and
it does not appear in the PES resulting from the CCSD(T) calculations when 6 = 90°.
When 6 = 90°, the CCSD(T) surface shows a continuous decrease in energy in the
BeF + H product channel, until the energy drops below the energy of the Be + HF
reactants. It is not until the angle 6 is below 90° that we begin to see this insertion
well beginning to form on the CCSD(T) PES. For the angles where the CCSD(T)
method correctly places the insertion well in the product channel of the Be + HF
reaction, the CCSD(T) method fails to correctly describe the region of the surface
where the Be—F and H—F bonds are stretched, as can be seen in Figures 11(a), 13(a),
and 15(a), corresponding to the CCSD(T) PES at 6 = 80°, 70°, and 45°, respectively.

For example, at 6 = 45° the CCSD(T) energies rapidly decrease as the Be—F distance

36

becomes greater than 6.0 bohr and the H—F distance becomes greater than 5.0 bohr,
until the energy actually falls below the energy of the reactants. This is not the case
for the CR—CCSD(T) or MRCI(Q) PESs. For the CR-CCSD(T) and MRCI(Q) PESs
at 6 = 45°, the energy increases continuously as the Be—F and H—F distances increase,
and both PESs stabilize at approximately 5.8 eV, as can be seen in Figures 15(b) and
15(c), respectively. Therefore, it is clear that the CR-CCSD(T) method is capable of
describing the deep insertion minimum corresponding to the formation of the HBeF
molecule, as well as all other regions of the PES, while the CCSD(T) method cannot
correctly describe the formation of the insertion minimum and provides an erratic
description of other regions of the PES.

As discussed earlier, when the Be—F—H angle 6 2 0°, there are two different
scenarios present: the Be atom inserted between the H and F atoms, and the Be
atom approaching the H atom of HF. Let us first look at the results for the case of
the insertion of the Be atom into the HF bond. The CCSD(T), CPI-CCSD(T), and
MRCI(Q) PESS for the insertion at 6 = 0° are shown in Figure 17. As one can see,
the CCSD(T) approach gives a qualitatively incorrect surface as the Be—F and Be—H
bonds become stretched. The maximum absolute errors in the CCSD(T) calculations
grow to be as large as 1.321 eV for R394: = 6.0 bohr and R394; 2 8.0 bohr, as seen
in Figure 18 and Table 4. For the CR-CCSD(T) method the results are much better.
The errors, relative to the MRCI(Q) results, are always below 0.4 eV for all Be—F and
Be—H distances. For most of the Be—F and Be—H distances at 6 = 0°, the errors do
not exceed 0.1—0.2 eV and are often in the range of 001—01 eV (see Table 4). In the

region of the deep insertion minimum, corresponding to the linear HBeF molecule, the

37

PESs calculated with the CCSD(T), CR-CCSD(T), and MRCI(Q) methods are all
very similar, as can be seen in Figure 17. The MRCI(Q) energy of the HBeF molecule,
at RBe—F = 2.59 bohr and R394, 2 2.49 bohr, is 3.98 eV below the energy of the
Be + HF reactants. This should be compared to the CCSD(T) energy at the HBeF
minimum, at R394: 2 2.59 bohr and R394, = 2.49 bohr, of —3.92 eV, relative to the
energy of reactants, and the CR—CCSD(T) energy at the HBeF minimum of —3.93 eV,
at RBe—F = 2.58 bohr and 1239-3 2 2.49 bohr. Even though the CCSD(T) energy at
the HBeF minimum is very close to the energy obtained in the MRCI(Q) calculations,
the topology of the CCSD(T) surface is incorrect. The CCSD(T) PES lies above the
MRCI(Q) surface near the HBeF minimum, but drops below the MRCI(Q) PES as
the Be—H bond is stretched (see Figure 18(a)). Meanwhile, the CR—CCSD(T) PES
lies only slightly above the MRCI(Q) PES (see Figure 18(b)). The PESs are virtually
parallel to each other.

In the case of 6 2 0°, corresponding to the Be atom approaching the H atom of
HF, the CCSD(T) approach once again gives unphysical results, whereas the CR-
CCSD(T) gives results which are close to the MRCI(Q) results. These results can be
seen in Figures 19 and 20, and in Table 5. The CCSD(T) method produces a PES
with an artificially low, pronounced barrier of ~3 eV near R384; 2 2.5 bohr and
R114: = 3.5 bohr, shown in Figure 19(a). The CR-CCSD(T) and MRCI(Q) PESs
show this barrier to be very flat and much higher in energy (see Figures 19(b) and
19(c)). As can be seen in Figure 20(a) and Table 5, the maximum absolute errors
in the CCSD(T) results, relative to the MRCI(Q) results, grow large as the Be—H

distance decreases and the H—F distance increases. The errors grow to as large as

38

2.773 eV for R3941 = 1.8 bohr and R34: 2 4.0 bohr. The errors for the CR-CCSD(T)
method are much smaller than those of the CCSD(T) approach. For most geometries,
the errors, relative to the MRCI(Q) results, for the CR—CCSD(T) method are on the
order of 001—01 eV. The largest absolute error, relative to the MRCI(Q) results, for
the CR-CCSD(T) method is only 0.428 eV, which can be seen in Table 5 and Figure
20(b).

Saddle-point geometries and energies were obtained for each angle 6 from 45°-
180°, with the CCSD(T), CR-CCSD(T), and MRCI(Q) methods and the cc-pVTZ
basis set. These results are compiled in Table 6. Figure 21 shows the differences
between the CCSD(T) and MRCI(Q), and the CR-CCSD(T) and MRCI(Q) energies
for the saddle point at each angle 6. It is easy to see from Table 6 and Figure 21 that
for larger values of 6 the saddle point energies obtained with the CCSD(T) method
are below those obtained with the MRCI(Q) approach, while for smaller values of 6
the opposite is true; the CCSD(T) energies are above the MRCI(Q) energies. For
example, at 6 2 180°, the CCSD(T) barrier energy is 1.30 eV, while the MRCI(Q)
barrier energy is 1.35 eV, but at 6 = 90°, the CCSD(T) barrier energy is 0.83, while
the MRCI(Q) barrier energy is 0.71 eV. The results of the CR-CCSD(T) saddle-
point energies show a different trend: the barrier energies are consistently slightly
above the barrier energies calculated with the MRCI(Q) method. The differences
between the CR-CCSD(T) and MRCI(Q) saddle—point energies range from 0.05 to
0.11 eV. This confirms the earlier observation that the PES calculated with the CR-
CCSD(T) method lies slightly above and parallel to the PES calculated with the

accurate MRCI(Q) approach, while the CCSD(T) method produces a PBS which is

39

not parallel to that of the MRCI(Q) method.

As mentioned earlier, we also used the CCSD(T), CR-CCSD(T), and MRCI(Q)
methods and the cc-pVQZ basis set to examine the BeFH system at 6 2 180°. The
results of these calculations can be seen in Figures 22 and 23 and Table 2. Once again,
as in the calculations for the BeF H sytem with 6 2 180° employing the cc-pVTZ basis
set, the CCSD(T) PES is qualitatively poor when compared to the MRCI(Q) PES.
At large Be—F and H—F separations, the CCSD(T) PES shows large errors relative to
the MRCI(Q) PES. The errors increase to a maximum of 4.077 eV and the surface
drops well below the MRCI(Q) surface. The CR-CCSD(T) PES, on the other hand,
is in nearly perfect agreement with the MRCI(Q) PES, as can be seen in Figures
22 and 23, and Table 2. The CR—CCSD(T) method is able to eliminate the failure
of the standard CCSD(T) method at larger Be—F and H—F separations. Also, the
product channel and the saddle-point energies of the CCSD(T) surface are too low
when compared to the same regions of the MRCI(Q) surface, while the CR-CCSD(T)
calculations produce a saddle point and product channel which are only slightly above
those resulting from the MRCI(Q) calculations. All of this can be seen by examining
the contour line at 1.3 eV in Figure 22 and noticing that the contour line at 0.0 eV
in Figure 22 appears in the product channel of the CCSD(T) PES, but not in the
product channel of the CR—CCSD(T) or MRCI(Q) PESs. The endothermicity of the
Be + HF —-) BeF + H reaction obtained with the CCSD(T) method and the cc-pVQZ
basis set is -0.051 eV. This value has the wrong sign when compared to the MRCI(Q)
value of 0.099 eV. On the other hand, the CR-CCSD(T) value for the endothermicity

of 0.301 eV retains the correct sign when compared to the MRCI(Q) value, and is

40

reasonably close to the experimentally derived value of 0.193 eV calculated from the

dissociation energies of BeF and HF.

4.5. Conclusion

The CR—CCSD(T) method has been shown to improve upon the failings of the
standard CCSD(T) method in the case of the Be + HF —> BeF + H, BeH + F, and
HBeF reaction, independent of the size of the basis set employed. The PES resulting
from calculations employing the CCSD(T) method has been shown to have the wrong
topology, especially in the product channel and in regions where the Be—F and H—F
bonds are stretched. Again, this finding is independent of the basis set employed.
The CR-CCSD(T) method produces a PES of the BeFH system which is virtually
parallel to and slightly above the full CI PES, in the case of the MIDI basis set, and
the MRCI(Q) PBS, in the case of the cc-pVTZ and cc—pVQZ basis sets.

The relatively low cost and ease-of-use of the CR-CCSD(T) approach make it an
attractive alternative to the considerably more expensive and complicated MRCI(Q)
approach. This seems to be particularly true for exchange and bond insertion reac-
tions such as those studied here, where single bonds are broken and formed, since
the CR—CCSD(T) method can produce results which are nearly identical to those

obtained with more expensive approaches such as MRCI(Q) with less time and effort.

41

5. Applications: The Vibrational Spectrum of trans-Imine
Peroxide (HNOO)

5.1. Background Information and Motivation

In December of 1998, the vibrational spectrum of the new molecule imine peroxide,
HNOO, was reported in articles in back-to—back issues of the Journal of the American
Chemical Society. The first was entitled “Laser Photolysis of Matrix-Isolated Methyl
Nitrate: Experimental and Theoretical Characterization of the Infrared Spectrum of
Imine Peroxide (HNOO)”, by Ling, Boldyrev, Simons, and Wight (LBSW).104 The
second was entitled “Reaction of NH (X) with Oxygen in a Solid Xenon Matrix:
Formation and Infrared Spectrum of Imine Peroxide, HNOO”, by Laursen, Grace,
DeKock, and Spronk (LGDS).105 Both studies used photolysis, matrix isolation, in-
frared spectroscopy, and electronic structure calculations. However, the assignments
of the fundamental frequencies in the infrared spectra to the vibrations in the HNOO
molecule differ greatly between the two studies. These differences can be seen in
Table 7. By examining Table 7, one can see that the major difference lies in the
assignments made to V3 and V4, which correspond to the NO and 00 stretching
motions, respectively. These frequencies are assigned at 1381.6 and 843.2 cm“1 by
LBSW, while LGDS assign them at 1092.3 and 1054.5 cm'l. According to Badger’s
rule,1°°‘108 the large difference in energy between V3 and V4, ~550 cm’l, from the
LBSW assignment, and the relatively small difference in energy between V3 and V4,
~50 cm“1, from the LGDS assignment, imply that there is a large difference in the

42

NO and CO bond lengths for the molecule assigned by LBSW, whereas there is a
small difference in these two bond lengths for the molecule reported by LGDS. If the
assignment of LBSW is correct, the large difference in the NO and 00 bond lengths
would make the HNOO molecule behave similar to the HONO molecule, while if the
LGDS assignments are correct the similar NO and CO bond lengths would mean
that the HNOO molecule would behave similar to ozone (and therefore have a partly
diradical character).

Obviously, the large discrepency between the LBSW and LGDS assignments needs
to be accounted for by a thorough theoretical examination with state-of-the-art com-
putational methods. Only one of the two interpretations (at most) can be correct.
Since HNOO is isoelectronic with ozone and HONO, and since both of these molecules
are implicated in atmospheric chemistry, it is important to determine whether HNOO
mimics one of these molecules. or if it is itself unique. Since both LBSW and LGDS
present evidence for only one isomer of HNOO, the trans-HNOO species, we focus on
the trans—HNOO species here.

In this section, we report the geometry, harmonic vibrational frequencies, and
anharmonic vibrational frequencies of trans-HNOO, obtained using a variety of CC
methods. These calculated vibrational frequencies are then compared to the results
published by LBSW and LGDS to determine which assignment of vibrational fre-
quencies is correct. The ability of high level CC methods to provide a very good de-

109‘1“ which is expected

scription of the harmonic frequencies of the ozone molecule,
to be similar to HNOO, makes CC methods an excellent choice for calculating the

highly accurate anharmonic force field necessary for computing fundamental vibra-

43

tional frequencies of the trans-HNOO molecule. Having the anharmonic frequencies
eliminates the risk of misinterpreting the spectrum by comparing the frequencies of
the fundamental vibrational transitions observed in experiments with the approxi-
mate vibrational frequencies resulting from oversimplified harmonic analysis.

The case of HNOO provides an excellent Opportunity to test the CR-CC methods,
which improve the results of standard CCSD(T) calculations in diradical systems. Al-
though HNOO cannot be a strong diradical, its similarity to ozone makes it a very
good candidate for a partly diradical system in which non-dynamical electron correla-
tion effects become important, since diradical systems can normally not be described
accurately by most single-reference methods. These effects are well described by the
CR-CC approaches, when diradical systems are examined. Thus, we expect good
results for the fundamental vibrational frequencies Of trans-HNOO from the CR—CC

calculations.

5.2. Computational Details

The geometry Of trans—HNOO was first optimized with the CCSD(T) approach
and the cc-pVTZ basis set, using the analytic gradient capability available in the
AGES 11 electronic structure package.112 Since high precision was necessary for the
calculation of the quartic force fields, very tight convergence thresholds were used for
the underlying RHF and CCSD calculations. The SCF density matrix in the RHF
equations was converged to 10‘12 and the CCSD equations for the maximum change in

cluster amplitudes defining the CCSD wave function were also converged to 10’”. All

44

the relevant cutoff thresholds for transformed integrals were set at 10‘20 or less, such
that all transformed integrals were kept, and the CCSD(T) geometry Optimization
was carried out until the RMS energy gradient was below 10‘10 hartree/bohr.

After the equilibrium geometry of trans-HNOO was determined with the CCSD(T)
method, a grid Of 263 nuclear geometries was generated from the CCSD(T) equilib-
rium geometry, for the subsequent anharmonic vibrational analysis. The energy at
each of the 263 nuclear geometries was calculated with the CCSD, CCSD(T), R-
CCSD(T), CR—CCSD(T), and CCSD(TQf), methods, with the same cc-pVTZ basis
set and the same tight convergence criteria that were used in the geometry optimiza-
tion.

To test the role of the iterative treatment of T3 clusters, additional energy cal-
culations were performed on a smaller subset of 32 nuclear geometries, pertaining
to the quadratic force field required for calculating harmonic frequencies, with the
CCSDT’3(Qf) method113 and the cc-pVTZ basis set. The CCSDT-3 approach, which
the CCSDT-3(Qf) method is built on, is an iterative triples method which provides
results of the full CCSDT quality, but with less computer cost. A non-iterative
quadruples (Qf) correction is then added to the CCSDT-3 energy to provide the final
CCSDT-3(Qf) energy. The harmonic frequencies calculated with the CCSDT-3(Qf)
approach were corrected for anharmonicity using the CCSD(TQf) anharmonicities.

The single point CCSD, CCSD(T), and CR—CCSD(T) calculations were per-
formed using the system Of CC computer programs developed by the Piecuch group,
which are available in the GAMESS electronic structure package. The CCSD(TQf)

and CCSDT-3(Qf) calculations were performed with the Michigan State Univer-

45

sity/ University of Silesia package of CC programs which is interfaced with the ACES
II Hartree—Fock and integral transformation routines.

The harmonic and quartic force fields were determined, with the help of Dr. Wes-
ley D. Allen from the University Of Georgia, using his INTDIF2003 computer code.114
The INDTIF2003 program is capable of computing any force field through sextic or-
der, for any molecule and any number of coordinates, with input from any order of
any analytic derivative method, including energy points alone. One does not have to
provide fully Optimized geometries to determine (an)harmonic force fields with IN T-
DIF2003. It is sufficient to provide the approximate geometry (the CCSD(T) one in
this case). INTDIF2003 contains central difference formulas through order 6, does
not resort to polynomial fitting, and has rigorous error control and monitoring. Nu-
merical errors in anharmonic frequencies can be reduced well below 0.5 cm'l, for the
tight wave function covergence used in this study. In the case Of our study, the har-
monic and quartic force fields for the trans-HNOO molecule were determined from the
available energy points, and the first numerical contamination did not appear until
6th order. A total of 262 displacements were necessary, in addition to the CCSD(T)
reference equilibrium geometry, with step sizes for the central difference computations
of 0.01 A for distances and 0.02 rad for angles, to determine the quartic force fields
for the CCSD, CCSD(T), CR—CCSD(T), and CCSD(TQg) methods.

As already pointed out, in the analysis of the quartic force field, the CCSD(T)
equilibrium geometry was used as a common equilibrium structure for all CC calcu-
lations and the quadratic, cubic and quartic force constants were calculated using the

CCSD(T) Optimized geometry as a reference geometry for each level of CC theory.

46

The key point is that at a fixed geometry the error in electronic structure computa-
tions diminishes as the order of the force constant increases, yet the higher-order force
constants are very sensitive to geometric perturbations. Accordingly, in the usual vi-
brational analysis scheme using the inaccurate equilibrium geometry, large errors in
the gradients at a given level of theory would cause a substantial loss of accuracy in
the calculation of higher-order force constants due to the insufficiently accurate equi-
librium structure. Thus, a first-order shift term has to be added to each PBS in order
to cancel out any spurious nonzero gradients that result from not using the Optimized
geometry, while leaving the more accurate higher-order force constants unchanged. In
this way, using a reference geometry, even though it may be nonstationary, can pro-
duce accurate anharmonic force fields which are devoid of errors in the corresponding
equilibrium structure. The details and theoretical basis for this have been developed
by Allen and Csaszar.115 After the determination Of the force fields was complete,
the vibrational anharmonic constants, vibration—rotation interaction constants, and
quartic and sextic centrifugal distortion constants were determined using vibrational
second-order perturbation theory, as applied to the vibration—rotation Hamiltonian
for semirigid asymmetric top moleculesm"119 Once the anharmonic force fields were
determined for all CC methods used in this study, the corresponding equilibrium ge-
ometries could easily be determined from the requirement that the energy gradient

at the equilibrium geometry is zero.

47

5.3. Standard and Completely Renormalized CCSD(T) and
CCSD(TQ) Calculations of the Geometry, Anharmonic Force Field,
and Vibrational Spectrum of trans-HNOO

The bond lengths and bond angles for the trans—HNOO molecule for each CC
method used in this study can be found in Table 8. The experimental and theoret-
ical results for the 121—126 fundamental frequencies of the trans—HNOO molecule are
listed in Table 9. As can be seen in Table 8, all CC methods predict the OO and
NO bond lengths to differ by only ~ 0.01 A, with both distances being in the range
of 1.28 and 1.31 A. This implies that the trans—HNOO molecule is more similar to
the ozone molecule than to the HONO molecule, both isoelectronic to it, since the
CO bond lengths in ozone are the same, while the NO bond lengths in HONO dif-
fer by approximately 0.25 A.12°:121 This, in turn, suggests that, just like ozone, the
trans—HNOO system has a partly diradical character and non-negligible contributions
from non-dynamical electron correlation effects. A balanced description of dynami-
cal and non-dynamical correlation effects requires the use of high-level CC methods,
such as CR-CCSD(T) or CCSDT-3(Qf). The CCSD(T) approach performs reason-
ably well when the diradical character is small but we expect to observe significant
improvements in the CCSD(T) results for the frequencies when the CR—CCSD(T)
and CCSDT-3(Qf) methods are employed. This is clearly seen in Table 9. The
CCSD(T) frequencies for trans—HNOO are reasonable, but the overall description of
these frequencies by the CR—CCSD(T) and CCSDT-3(Qf) approaches is clearly much
better. In fact, the CR—CCSD(T) approach significantly improves on the description

of the 14-44, frequency difference by the CCSD(T) method, providing a result which

48

is virtually identical to that obtained in the LGDS assignment. This confirms our
expectation that the non-dynamical electron correlation effects, which are not well
described by CCSD(T), and which CR-CCSD(T) treats in an accurate manner, are
quite significant in trans—HNOO.

The results for the fundamental frequencies reported in Table 9 strongly favor
the assignment of LGDS. If we focus on the NO stretching frequency and the OO
stretching frequnecy, V3 and V4, respectively, in Table 9, it can be seen that the
theoretical calculations are much closer to the assignment of LGDS and far from
the assignment in the LBSW study. The LBSW assignment for the NO stretch,
V3, is 1381.6 cm‘l, whereas the assignment of this vibrational band from the LGDS
work is 1092.3 cm‘l. These experimental assignments should be compared to the
theoretical frequencies from the CC calculations which produce energies for V3 of
1147, 1116, 1123, and 1126 cm“1, for the CCSD(T), CR—CCSD(T), CCSD(TQf),
and CCSDT'3(Q{) methods, respectively. For the OO stretching frequency, V4, the
LBSW assignment is 843.2 cm‘I, while the LGDS assignment is 1054.5 cm‘l. This
should be compared to the theoretical values of 1042, 1078, 1047, and 1071 cm‘l, for
the CCSD(T), CR—CCSD(T), CCSD(TQf), and CCSDT-3(Qf) methods, respectively.
The excellent performance of the CR-CCSD(T) approach, which gives small errors
relative to the LGDS data, on the order of 20 cm‘1 for these two frequencies, should
be pointed out.

It is clear from Table 9 that the CC predictions for V3 and V4 agree much better
with the LGDS assignment than with the assignment proposed by LBSW. This is

also true for the other bands in the experimental spectra. Looking at the values for

49

V1, the NH stretching frequency, in Table 9, the calculated value Of ~ 3190 cm“1
is close to the LGDS value of 3165.5 cm‘l, while the LBSW frequency for this NH
stretch is far removed from the theoretical value. The second band, characterizing the
HNO bend, was not observed in the assignment of LBSW. LGDS report this band
at 1485.5 cm‘l, which is very close to the theoretical value of ~ 1500 cm’l. The
fifth band, which pertains to the NOD bend, was not Observed in the spectrum of
LGDS, whereas LBSW report it at 670.1 cm‘l. The calculated value is ~ 650 cm’l.
Finally, the torsion frequency, V6, was reported by LBSW at 790.7 cm“1. The LGDS

assignment is 764.0 cm‘1

, in excellent agreement with the CC values in Table 9.
Now we turn to the isotopic shift data reported by LBSW and LGDS. The work
of LBSW included deuterium substitution, while the LGDS work included 180 sub-
stitution. The experimental and theoretical isotopic shift data are summarized in
Table 10. In the deuterium substitution work Of LBSW, there was an observed shift
Of 844 cm’1 in the first band and an observed shift of 301 cm“1 in the third V3.
The agreement with the calculated theoretical shifts is quite poor for V3. Indeed the
theoretical shift of the first band is approximately 800 cm‘1 and the theoretical shift
of the third band is ~ 10 cm‘l, in poor agreement with the LBSW data. The fourth,
fifth and sixth isotopic shifts assigned by LBSW show an equally poor agreement with
computed shifts, as can be seen in Table 10. The LBSW assignment puts these shifts
at 20, 11, and 203 cm‘l, respectively, while the computed theoretical shifts for these
three bands are ~ 80, ~ 20, and ~ 180 cm‘l, respectively. This shows, once again,

that the assignment reported by LBSW is incorrect. This should be contrasted with

the 18O isotopic substitution data of LGDS, which perfectly agrees with our theoreti-

50

cal data (agreements to within ~ 1 cm‘l). The isotopic shift data in Table 10 provide
further evidence that the assignment of LGDS is correct, while that of LBSW is not.

The techniques utilized by both experimental groups to obtain the spectrum Of
trans—HNOO were similar. Both groups used photolysis, matrix isolation, and in-
frared spectroscopy. Thus, one may wonder what went wrong in the LBSW experi-
ment or the subsequent vibrational analysis that caused the LBSW group to provide
an entirely incorrect interpretation of their spectra. One of the factors is the fact that
the reactions used to create the HNOO molecule by the LBSW and LGDS groups were
very different. The LBSW work used the decomposition of methyl nitrate. LBSW
proposed two decomposition channels of CH3N02. One channel forms formaldehyde
and hydrogen nitrile (HN02), and the other forms formaldehyde and imine peroxide.
This decomposition reaction produces a plethora of bands in the infrared spectrum,
which can be interpreted in many ways. Out of the 23 bands that were found by
LBSW after 20 minutes of photolysis, and 41 bands after 520 minutes Of photolysis,
5 were eventually assigned to trans—HNOO. A strong factor in this assignment was
the theoretical work using the widely criticized QCISD122 method, which in overview
should never be used in good calculations. In particular, this method does not have
the ability to accurately describe the HNOO molecule, as will be shown later.

In the work of LGDS, hydrazoic acid (HN3) was photolyzed to HN and N2, and
then HN was reacted with Oz to form HNOO. This procedure is simple and very clean.
It produces impurity molecules, such as HONO, NHgOH, N3H2, H20, and C03, but
these impurities could be easily identified based on other matrix isolation experiments,

and the assignment of the new bands to HNOO was based on the relatively simple

51

chemistry of the system.

Since HNOO is isoelectronic with both ozone and HONO, it is interesting to
compare the vibrational stretching frequencies of HNOO with each of these molecules.
The assignment by LGDS of the NO and 00 stretching frequencies to 1092.3 and
1054.5 cm‘1 are very close to the stretching frequencies in ozone of 1103 and 1042
cm‘l.123 The separation Of these bands in HNOO is 37.8 cm‘l, which is very close to
the difference in frequency for the ozone stretches of 61 cm‘l. The LBSW assignments
of these two bands in HNOO at 1382 and 843 cm‘l, for a separation of 539 cm‘l,
are far removed from the corresponding bands of ozone. In the case of trans—HONO,

the experimental NO stretching frequencies are 1680 and 794 cm’1,124'125

giving a
separation of 886 cm“. This splitting is similar to the splittings from the assignment
of LBSW to HNOO, but not the splitting obtained from the assignment Of LGDS. This
means that the LGDS assignment suggests that HNOO has an electronic structure
similar to that of ozone, while the LBSW assignment suggests that HNOO has an
electronic structure similar to HONO. Our accurate CC-Optimized geometries in Table
8 clearly suggest that trans—HNOO is similar to ozone.

As mentioned earlier, LBSW used the QCISD theoretical method to aid in the as-
signment of the experimental bands to the trans—HNOO molecule. This is important
to note since the QCISD method produces wildly varying degrees of accuracy when
applied to calculations of HONO and ozone, both of which are isoelectronic with
HNOO. For the “easy”, single—reference, closed-shell HONO molecule, the QCISD
method performs reasonably well for both bond distances and stretching frequen-

cies.1°“ However, when the QCISD method is applied to the much more challenging

52

ozone molecule, the results are very poor. The QCISD method predicts a separation
of 258 cm”1 for the stretching frequencies in ozone, which is quite different from the
experimental value of 61 cm‘l. This is due to the multi-reference diradical character
Of the ozone molecule, which means that it is necessary to include connected triply
and, perhaps, even quadruply excited clusters in an accurate manner to obtain high
quality results for the ozone molecule. The CC methods used in this study, such as
CR—CCSD(T) and CCSDT-3(Qf), include these clusters and can produce an accu-
rate description of both HONO and ozone. We can clearly see this by comparing
the results of our highly accurate vibrational analysis, based on the CR-CCSD(T),

CCSDT-3(Qf), and other CC calculations, with the LGDS data, as discussed above.

5.4. Conclusion

The CCSD(T), CR—CCSD(T), CCSD(TQf), and CCSDT-3(Qf) methods have
been used to calculate the anharmonic frequencies of trans—HNOO. All of the CC
methOds, particularly the CR-CCSD(T) and CCSDT’3(Q{) approaches, produce NO
and 00 bond lengths which are nearly identical, and NO and 00 stretching frequen-
cies Of approximately 1120 and 1070 cm’l, respectively. These results compare very
well with the NO and 00 stretching bands assignment of LGDS of 1092 and 1055
cm“, respectively. This clearly suggests that trans—HNOO was formed and properly
identified in the LGDS experiments.

The experimental results of LBSW do not seem to provide compelling evidence for

the formation of HNOO from the photolysis of CH3N02. This is due to the complexity

53

of the photolysis, resulting in too many photo-products in the matrix isolation study.
The apparent inability Of the QCISD method, which was used by LBSW to aid in
the assignment of the vibrational bands, to accurately describe the HNOO molecule,
clearly contributed to the failure of the analysis reported by LBSW.

The results of our theoretical work strongly favor the assignment of the vibra-
tional bands of trans—HNOO by LGDS over that of LBSW. They also demonstrate
an ability of the CR-CCSD(T) method to resolve controversies in experimental work,
particularly when the molecular species in question has a more complicated elec-
tronic structure, as is the case when trans—HNOO is examined. The trans—HNOO
system seems very similar to the challenging ozone molecule, which requires a high-
level treatment Of electron correlation effects. We demonstrated here that the CR-
CCSD(T) approach can provide such treatment in the demanding situation created

by the ozone-like trans—HNOO species.

54

6. Applications: The Mechanisms Of the Cope Rearrangement

of 1,5-Hexadiene and the Bergman Cyclizations of Enediynes

6.1. Background Information and Motivation

As mentioned in the Introduction, single-reference approaches have difficulty de-
scribing systems which have a significant amount of diradical character. In fact, the
proper description of the reaction mechanisms involving diradical species is a chal-
lenge to the vast majority Of electronic structure methods, including even the best
multi-reference approaches (cf,, e.g.. References 95 and 126). The main challenge
is to accurately balance the diradical and closed-shell regions of the PES, and dy-
namical and non-dynamical electron correlation effects, which is a problem for many
approaches.

The difficulties in balancing dynamical and non-dynamical correlation in elec-
tronic structure calculations involving diradical species resulted in controversy over
the mechanism of the Cope rearrangement of 1,5-hexadiene.l27’133 Experimental in-
vestigations ruled out the dissociative mechanism involving a bis-allyl structure, but
it is much more difficult to decide between the two other alternatives: (i) a concerted
a bond shift involving an aromatic transition state (TS) or (ii) a two-stage process
involving a stable 1,4-diyl diradical intermediate (see Figure 24).

Experimental studies alone may not be conclusive about the exact nature of the

Cope rearrangement of 1,5-hexadiene. Thus, support from theory is necessary to

55

discern the true mechanism Of this reaction. Unfortunately, the results of theoretical
studies of the Cope rearrangement of 1,5-hexadiene strongly depend on the method
employed in the calculations. For instance, RHF, CI, and some DFT calculations sup-
port a mechanism involving an aromatic TS, whereas semiempirical and second-order
Moller—Plesset (MP2) calculations result in a concerted pathway through a diradi-
cal cyclohexane-1,4-diyl intermediate.127 Some other DFT and CASSCF calculations
produce results which make both pathways simultaneously viable.127‘130 The highest
and most reliable level of theory that had previously been used to study the Cope
rearrangement Of 1,5-hexadiene was multi-reference perturbation theoryms‘130 Two
different versions of this theory, the MROPT2 approach Of Kozlowski and David-
son,134 and the CASPT2 method of Roos et al.,135'13° predict a single aromatic TS
on the PES,128‘13° which is in agreement with experimental findings that support the

137 Based on these multi-reference perturbation theory calcula-

concerted mechanism.
tions, the Cope rearrangement of 1,5-hexadiene involves an aromatic TS which has a
chair conformation lying along a Cg), cut of the PES defined by the interallylic distance
R, as shown in Figure 24. This TS, which represents a minimum energy structure on
the Cg), cut Of the PES, was determined to be aromatic by Staroverov and Davidson,
who analyzed it with the concept of the density of effectively unpaired electrons.128'138
Other calculations have provided similar conclusions.139* 14° The concerted mechanism
is also supported by good agreement between the secondary kinetic isotope effects ob-
tained with DFT for loose TS structures and experiment. Agreement between the
secondary kinetic isotope effects and experiment is significantly worse when the tight

1,4-diyl-like intermediates are assumed in the calculationsm'142

56

Although the analyses provided by the previous work127‘130 are quite convincing,
it is essential to reexamine the Cope rearrangement Of 1,5-hexadiene with meth-
ods which can provide a highly accurate description of dynamical correlation effects
(MROPT2 and CASPT2 offer a low-order treatment of dynamical correlation) and
which can balance dynamical and non-dynamical electron correlation effects in di-
radical, aromatic, and closed-shell systems. The effects of dynamical correlation can
be seen by comparing the results of CASSCF calculations, which describe only non-
dynamical correlation effects, and the results of the CASSCF based multi-reference
perturbation theory calculations, which describe both non-dynamical and dynamical
correlation effects. In the case of the CASSCF results, two minima are found on
the Cg), cut of the PES, one in the diradical region and one in the aromatic region,
while for the results of the multi-reference perturbation theory calculations only a
single aromatic TS is predicted on the Cg), cut (see References 128—130 and Figure
26 in Section 6.3). It is generally acknowledged that dynamical correlation effects are
most accurately described by CC theory. However, standard CC approaches, such
as CCSD(T) (sometimes regarded as the “gold-standard” of electronic structure the-
ory), fail to correctly describe the Cope rearrangement of 1,5-hexadiene. As shown in
Section 6.3, the CCSD(T) method produces a reaction pathway through a diradical
structure or produces two nearly isoenergetic minima on the Cg), cut of the PES, both
of which are shifted toward the diradical region of the PES. This is due to the artifi-
cial lowering of the energies of the diradical-structures, relative to the aromatic and
closed-shell structures, by the CCSD(T) method. As shown in the previous sections,
the CR-CCSD(T) method can accurately describe PESS which involve bond breaking

57

and diradical structures. Thus, we expect a good performance of the CR—CCSD(T)
approach in calculations involving the C3,, cut of the PES of the Cope rearrangement
of 1,5-hexadiene. In section 6.3, we present the results for the Cope rearrangement
of 1,5-hexadiene obtained with the CR—CCSD(T) method, and compare those results
with the results of CCSD, CCSD(T), DFT, and multi-reference perturbation the-
ory calculations. We also show in section 6.3 how the CR-CCSD(T) approach can
provide useful insights into the degree Of diradical character Of the TS of the Cope
rearrangement of 1,5-hexadiene.

Another challenging problem for standard CC approaches is the Bergman cycliza-
tion of enediyne molecules. In the 1990’s a group of antibiotics was discovered which
contained enediyne groupsm“152 These antibiotics incorporate the enediyne unit into
nine— or ten—membered ring structures and possess the ability to destroy the DNA
Of bacteria and viruses, as well as cancerous tumor cells. The molecule containing
the enediyne ring structure docks into the minor groove of the DNA, triggering a
Bergman cyclization of the enediyne ring that results in the formation of a diradical.
This diradical then abstracts two hydrogen atoms, one from each strand of DNA, to
form a stable arene structure and a DNA radical. This causes a cleavage of the DNA
and leads to the death of the cell.

The discovery and development of new anticancer drugs which contain enediyne
groups is a long and costly process. Clearly, the time and money spent to find these
new drugs can be significantly lessened by the use of computational chemistry. Kraka
and Cremer have mapped out the requirements necessary for the structural features

of any enediyne compounds which could be possible candidates as new anticancer

58

drugs.153 According to these authors, the energetics of the Bergman cyclization re-
action is very important to the likelihood that an enediyne molecule would be an
acceptable candidate for an anticancer antibiotic. An accurate description of the
reaction pathway leading to the diradical product molecule is very important in de-
scribing the H-abstraction ability of the diradical as well as its kinetic stability.153

In Section 6.4, we show the results of the CR-CCSD(T) and CR-CCSD(TQ) calcu-
lations for the enediyne structures which could be possible candidates to be incorpo-
rated intO anticancer medications, as suggested by Kraka and Cremer.153 The CR-CC
results are compared to the results of DFT, CCSD(T), and CCSD(TQf) calculations.
In many cases, the standard CC approaches drastically underestimate the energy of
the diradical product Of cyclization, relative to the reactant molecule, and thus give
an incorrect estimate of the viability of these structures as possible antitumor agents.

The CR—CC methods are shown to be more reliable when calculating the cyclization

reaction pathway.

6.2. Computational Details

To examine the mechanism Of the Cope rearrangement of 1,5-hexadiene, the Cg),
cuts of the PES along the coordinate describing the interallylic distance R were cal-
culated. The Cg), cuts were calculated with the CCSD, CCSD(T), CR-CCSD(T),
DFT (using the UB3LYP hybrid functionall54‘15°), CASSCF, and second-order multi-
reference perturbation theory (the MCQDPT method of Nakanol57'158). For each

method, the Cg), cut was calculated with two basis sets, 6-31G"‘159 and 6-311G""",160

59

to show that the main conclusions are not basis set dependent. Following the earlier
work of Davidson et al., the geometries defining the Cg), cut were Obtained by Opti-
mizing the structures at fixed values of the interallylic distance R, over a range of
1.5 to 4.0 A, using the UB3LYP functional as implemented in Gaussian 98,161 with
each basis set. These Optimized structures were then used to calculate single point
energies using the CCSD, CCSD(T), CR-CCSD(T), CASSCF, and MCQDPT meth-
ods. There were 41 densely spaced values of R considered for calculations involving
the 6-31G* basis and 37 densely spaced values of R considered for the calculations
employing the 6-311G** basis set. All CCSD, CCSD(T), CR—CCSD(T), CASSCF,
and MCQDPT calculations were performed with the GAMESS computer package of
electronic structure programs (the CC routines in GAMESS have been provided by
the Piecuch group). The active space for the CASSCF and MCQDPT calculations
employed 6 active electrons and 6 active orbitals. The active orbitals chosen were
the 7ag, 5a“, 7b“, Sag, 5bg, and 8bu orbitals. This orbital choice includes all orbitals
necessary to describe the migrating 7r and or bonds in the Cope rearrangement reac-
tion of 1,5-hexadiene. This choice also ensures that the CASSCF reference function
includes the |~--7a§5afi7bfi|, |~--7a§5a,2,8a§|, and |---7a§5b§7bfi| electron configura-
tions, which must be included to provide a balanced description of both the aromatic
and diradical regions of the PES. In regions of intermediate interallylic distances R,
the 02}; 1,5-hexadiene molecule is aromatic and the closed shell |--~7a§5afi7bfi| de-
terminant dominates the wave function, but this description is no longer adequate at
long and short interallylic distances R. At short values of the interallylic distance,

the 1,5-hexadiene molecule becomes a diradical 1,4-diyl with the wave function

60

‘I’din = I ~--7a§5afi7bfi| — c(7b,2l —> 8(12) I - - - 7a§5a38a§| + - . -, (43)

where the coefficient c(7b,2l —) 8a:) of the I---7a§5a38a§I configuration, which is a
double excitation from the I - - - 7a§5afi7bfi| determinant, approaches 1 as the interal-

lylic distance R decreases. At longer interallylic distances, the molecule becomes a

complex of two allyl radical molecules with the diradical wavefunction

\Ilbis_auy1 = I - - -7a§5afi7bfi| — C(5afi —) 5b,?) I - - - 7a§5b§7bil + - ~ -, (44)
where the cofficient C(5afi —> 5b:) of the I---7a§5b§7bfiI configuration, which can
be Obtained by a 5a,2, —) 5b: double excitation from the I~-7a§5afi7bfil determi-

nant, becomes 1 in the limit of infinite interallylic distance R. The two coeffi-
cients, c(7b,2l —> 8ag) and c(5a,2l —+ 5b:), in the above expressions correspond to
the doubly excited cluster amplitudes t(7bfi —> 8ag) and t(5a,2l —> 5bg), respectively,
which can be Obtained from the CCSD or other CC calculations. In the exact case,
t(7bfl ——> 8a§) = —c(7b,2, —+ 8ag) and t(5a:"1 —) 5b:) = —c(5afi —+ 5b:). It is shown later
that these cluster amplitudes correlate with the size of the denominator DIT) in the
CR—CCSD(T) energy expression (cf. Equation (30)). Thus, the value of D”) can be
used to provide insight into the amount of diradical character a molecule possesses.
The reactions used for the study of the Bergman cyclization of enediyne molecules
are shown in Figure 25. The reaction shown as System 1 in Figure 25 has been studied
by many authorsl"""1‘52“173 using a variety Of ab initio methods. As in the case of the

Cope rearrangement of 1,5-hexadiene, the main problem with calculations of reactions

61

shown in Figure 25, which all involve diradical species, is balancing the non-dynamical
and dynamical correlation effects along the reaction pathways. Thus, the reactions
shown in Figure 25 are good examples for testing the CR-CC approaches.

In order to investigate the reactions shown in Figure 25, it was first necesary to
optimize the geometries of the reactant, TS, and product molecules. Such calcula-
tions for the relatively large molecules in this study are too costly for CC methods.
Thus, a less expensive alternative was chosen for the optimizations. Investigations
by Grafenstein and coworkers174 have shown that the DFT approach can provide
reasonably accurate results for Bergman cyclization reactions, as long as one uses
unrestricted DFT (UDF T) in calculations of the diradical structures. Thus, the ge-
ometries of the enediyne structures in this study were Optimized at the DFT level
using the B3LYP functional and the 6-31G** basis set following the procedure of
Kraka and Cremer.153 The unrestricted B3LYP (UB3LYP) approach was used for
the diradical product molecules, whereas the restricted B3LYP (RB3LYP) approach
was used for the reactant and TS structures. These Optimized geometries were then
used to calculate single-point energies using the CCSD, CCSD(T), CR—CCSD(T),
CCSD(TQf), and CR-CCSD(TQ) methods with the same 6-31G** basis set. The
CC codes developed by Piecuch et. a1. and available in the GAMESS computer
package were used for all CC calculations, while Gaussian 98 was used for all DFT

calculations.

62

6.3. Results for the Mechanism of the Cope Rearrangement of
1,5-Hexadiene

The Cg), cuts of the PBS for the Cope rearrangement of 1,5-hexadiene calculated
with the UB3LYP, CASSCF, MCQDPT, CCSD(T), and CR-CCSD(T) methods can
be found in Figure 26. The Cg), cuts Of the PES resulting from the UB3LYP and
CASSCF methods with the 6-31G* basis set show two minima, as can be seen in
Figure 26(a). The UBBLYP calculations show a deep minimum in the aromatic region
Of the surface at R = 1.97 A, and a shallow minimum in the diradical region of the
PES at R = 1.65 A. For the CASSCF method, the deep minimum is in the diradical
region around R = 1.64 A and the shallow minimum is in the aromatic region of the
Cg), cut near R = 2.2 A. The CASSCF/6-31G* method shows a greater propensity for
the stepwise mechanism through the diradical intermediate due to the existence of the
deeper minimum on the Cg), PES cut in the diradical region, while the UB3LYP/6-
31G* method favors the concerted mechanism involving an aromatic TS which is
lower in energy, than the other minimum of the UB3LYP curve shown in Figure
26(a). When the larger 6-311G** basis set is used, the UB3LYP and CASSCF results
are altered, but neither of the two curves resulting from the UB3LYP/6-311G**
and CASSCF/6-311G** calculations seems correct. As shown in Figure 26(b), the
UB3LYP/6-311G** method produces only one pronounced minimum in the aromatic
region of the PES near R 2: 2.04 A, but there remains a significant change in curvature

in the diradical region at shorter interallylic distances, which closely resembles the

shallow minimum on the UBBLYP/6-31G* PBS at R = 1.65 A. In the case of the

63

CASSCF/6-311G** method, the PES of the Cg), cut still has two minima, although
there are differences when this curve is compared to the curve produced with the
CASSCF/6-31G* calculations. The CASSCF/6-311G** calculations still produce a
minimum in the diradical region at R = 1.64 A, but the second minimum is now
shifted toward the bis-allyl region of the surface near R = 2.35 A. Although the
energy difference Of 0.6 kcal/mol between the two minima is smaller in the CASSCF/6-
311G** case, when compared to the CASSCF/6-31G* curve (with the minimum at
R = 1.64 A now higher in energy when the larger basis set is employed), the overall
shape of the curve remains incorrect.

The Cg), cuts produced from the higher level theoretical methods, shown in Figure
26, are much different from those produced by the UB3LYP and CASSCF methods.
The results of the CCSD(T), CR-CCSD(T), and MCQDPT approaches produce Cg),
cuts which have only a single minimum when the 6-31G* basis set is employed.
There are, however, differences between the results of the CCSD(T) calculations,
and the PESs resulting from the CR-CCSD(T) and MCQDPT calculations. The
CR—CCSD(T) and MCQDPT approaches have minima in the aromatic region of the
surface, at R = 1.83 A and R = 1.86 A, respectively, while the minimum on the
CCSD(T) PBS at R = 1.72 A, is significantly shifted toward the diradical region,
as can be seen in Figure 26(a). This situation is different when the larger 6-311G**
basis set is employed (see Figure 26(b)). The CR-CCSD(T) and MCQDPT Cg), cuts
are still very similar. Each has a single minimum in the aromatic region, at R =
1.86 A when the CR—CCSD(T) method is employed and at R = 1.88 A, when the
MCQDPT approach is used. The CCSD(T)/6-311G** curve has now two minima,

64

one closer to the aromatic region at R = 1.82 A and another one in the diradical
region at R = 1.72 A. These two minima on the CCSD(T)/6-311G** PES cut are
nearly isoenergetic and are separated by a very small 0.2 kcal/mol barrier. In this way,
the CCSD(T) approach makes the stepwise pathway through the 1,4-diyl diradical
intermediate more accessible than either the CR—CCSD(T) or the MCQDPT methods.
It is easy to see from Figure 26, that the CR-CCSD(T) method removes the failure
of the standard CCSD(T) approach, producing a single minimum, like the MCQDPT
method, which is in close proximity to the minimum on the MCQDPT PES cut.
The CCSD(T) method produces minima which are far removed from the MCQDPT
minimum (two minima in the case of the 6-311G** basis set).

The activation energies, AEI, corresponding to the minima on the PESs in Figure
26, are shown in Table 11. The activation energy calculated from the CCSD(T) re-
sults would seem to indicate that the method provides a good description of the Cope
rearrangement of 1,5-hexadiene, with AB1 = 36.2 and 35.2 kcal/mol for the 6-31G*
and 6-311G** basis sets, respectively. These values are close to the experimentally
derived value Of 35.0 kcal/mol reported by Davidson, et al.127 This excellent agree-
ment, however, between the CCSD(T) and experimentally derived AE1 value is, most
likely, due to a fortuitous cancellation Of errors, since the structures corresponding
to minima on the CCSD(T) 02,, cut are in the diradical region of the surface, which
contradicts the accepted interpretation that the Cope rearrangement of 1,5-hexadiene
is a concerted process through an aromatic transition state. Based on the results in
Table 11, we can expect that if the size of the basis set were increased even further,

the value of AE1 calculated with CCSD(T) would drop below that Of the experimen-

65

tally derived result. This should be contrasted with the AE1 values obtained with the
CR—CCSD(T) approach, which are the upper bounds to the experimentally derived
AEI value. The CR-CCSD(T) method produces activation energies of 38.9 and 37.7
kcal/mol for the 6~31G* and 6-311G** basis sets, respectively, and it should be noted
that these energies correspond to aromatic transition state structures defining the
concerted mechanism. Based on the observed patterns, it is expected that further in-
crease of the basis set size would bring the resulting AEt value for the CR-CCSD(T)
method closer to the experiment. One should also notice from Table 11 that the ac-
tivation energies calculated with the MCQDPT approach are considerably below the
experimentally derived .AEI value. This might be a consequence of the inability of the
multi-reference perturbation theory methods to provide a well balanced description
of both transition state and reactant/ product structures.95'12°
As shown, for example, in Refs. 43-45, it is the denominator Dm in the triples
correction of the CR-CCSD(T) energy expression which improves the CCSD(T) re-
sults in the diradical/ bond breaking regions. It is interesting to examine the role of
D”) here. It is also interesting to examine the relationship between the magnitude
of the denominator DIT) and the t(7b,21 ——> 8ag) and t(5a,2, —) 5bg) cluster ampli-
tudes which, as explained earlier, measure the significance Of the I - - ~7a§5afi8a§| and
I---7a§5b§7b,2,| configurations, respectively, in the wave function. Since this relation-
ship is almost independent of the size of the basis set, we will focus only on the smaller
6-31G* basis set results.
I At R = 1.83 A, the point for which the CR-CCSD(T)/6-31G* method produces a

minimum on the Cg), cut of the PES, the value of the denominator D”) is 1.37. This is

66

very close to the Dm = 1.35 value Obtained for the closed-shell 1.5-hexadiene reactant
molecule. At R = 1.72 A, which is the interallylic distance for which the CCSD(T)/6-
31G* method produces a minimum on the Cg), cut, Dm = 1.40. There is an increase
in the value of D”), when going from R = 1.83 A to R = 1.72 A, relative to the closed-
shell reactant molecule, which is related to the increase in the diradical character of
the wave function as the interallylic distance is shortened. The larger value of DIT) at
R = 1.72 A means that the CCSD(T) approach produces a transition state with more
diradical character than the CR-CCSD(T) method. This can be shown by examining
the absolute value of the t(7bf’1 —) 8ag) cluster amplitude, which determines the 1,4-
diyl diradical character of the wave function. The value of the t(7b,2Jl —+ Sag) cluster
amplitude resulting from the CCSD/6-31G* calculations increases from 0.16 at the
CR—CCSD(T) minimum of R = 1.83 A, to 0.22 at the CCSD(T) minimum of R = 1.72
A. The values of DIT) and t(7bfi —) 8ag) are also relatively large for the R values that
correspond to the diradical minima on the UB3LYP/6-31G* and CASSCF/6-31G*
surfaces. For instance, the value Of the denominator D(T) at the UB3LYP/6-31G*
minimum at R = 1.65 A is 1.43, which is significantly higher than the Dm value
of 1.35 Obtained for the closed-shell reactant. The absolute value of the t(7b3 —> Sag)
cluster amplitude resulting from the CCSD/6-31G* calculations of 0.28 is also quite
large at R = 1.65 A. At the same time, the value of Dm corresponding to the aromatic
minimum on the UB3LYP/ 6-31G* Cg), cut at R = 1.97 A, of 1.37, is very close to the
Dm value for the closed-shell reactant. This nicely correlates with the small value

of the t(7bfi —-> Sag) cluster amplitude, at R = 1.97 A, is only 0.12. Likewise, the

67

absolute value of the t(5a,2l —-) 5bg) cluster amplitude, which describes the amount of
bis-allyl diradical character, calculated at the UB3LYP/6-31G* aromatic minimum,
is also relatively small (0.07).

The dependence of the denominator DIT), and the t(7bfi —> 8a,?) and t(5a,2l —> 51):)
cluster amplitudes, as obtained in the CR-CCSD(T)/6-31G* calculations, on the in-
terallyic distance R are shown in Figure 27. As one can see, the denominator Dm is
large in the region Of tight 1,4-diyl diradical structures (R S 1.7 A), it passes through
a minimum in the region of aromatic structures (R z 2.0 A) to increase again as
the molecule becomes two separate allyl radicals (R > 3.0 A). By comparing Figures
26(a) and (b), one can see that there is a correlation between the-value of the denom-
inator DIT) and the values Of the t(7b,2l —) 8ag) and t(5a,2l ——> 51):) cluster amplitudes.
The absolute value Of the t(7bfi —> 8ag) amplitude, which tells how important the
Iu-7ag5a38agl configuration is in the wave function, increases from 0.05 R = 4.0 A
(the bis-allyl region), to 0.11 at R = 2.0 A (the aromatic region), and to 0.50 at
R = 1.5 A (the 1,4-diyl region Of the Cg), cut of the PES). The absolute value of the
t(5a,2l —> 51):) cluster amplitude, which determines how important the I - - - 7a§5b§7bfi|
configuration is in the wave function, shows the Opposite trend. It decreases from 0.59
at R = 4.0 A, to 0.08 at R = 2.0 A, and, finally, to 0.01 in the 1,4-diyl region of the
surface, at R = 1.5 A. The curves representing the dependence Of the t(7b,2, —> 8a;‘;) and
t(5a,2l —> 5bg) cluster amplitudes cross at R z 2.1 A, with t(7b,2, —> 8a,?) = t(5a,2l —> 517:)
= —0.09. When this happens the I---7a§5afi8a§| and I---7a§5b§7bfi| configurations

are of equal importance in the wavefunction. This point Of equal significance of the

68

I - - - 7a§5afi8a§| and | - - -7a§5b§7bfi| configurations is in the aromatic region of the sur-
face and is near the point where the denominator DIT) approaches its minimum value
of 1.37, at R z 2.0 A. As the value of R becomes smaller, corresponding to the tight
1,4-diyl like structures, or larger, corresponding to the loose bis-allyl like structures,
the diradical character Of the molecule increases, as does either the t(7b,2l ——> 8ag) or
t(5a,2l —> 51):) amplitude. This, in turn, increases the value of the denominator DIT).
This is a consequence of the definition of the denominator Dm in terms of the clus-
ter amplitudes obtained from the CCSD method, Equation (34). As we can see now,
Equation (34) defining DIT) brings important chemistry into the CR—CCSD(T) cal-
culations which is missing in the CCSD(T) calculations, since Dm = 1. As the wave
function increases in its diradical character, the absolute value of the t(7b§ —> Sag)
or t(5a,2, —> 5bg) amplitude increases, causing the standard CCSD(T) approach to fail
by producing an unphysically large triples correction. In the CR-CCSD(T) method,
the value of the denominator DIT) increases as well, damping the large triples energy
correction and improving the CCSD(T) results in the diradical regions of the PES.
Since the CR—CCSD(T) approach provides a correct description of the Cg), cut
of the PES by sealing down the triples correction of CCSD(T), one may wonder if
the CCSD method itself provides the correct description. As shown in Figure 28 and
Table 11, the CCSD method does not provide accurate results. It is true that the
minima on the CCSD Cg), PES cuts obtained with the 6—31G* and 6-311G** basis sets
are located at R = 1.87 A and R = 1.89 A, respectively, so they are very close to the
minima on the CR-CCSD(T) and MCQDPT curves. However, the activation energies

of 42.2 kcal/mol and 41.1 kcal/mol, calculated using CCSD, are considerably higher

69

than the CR—CCSD(T) and experimentally derived values, as can be seen in Table
11. Also, the CCSD energies in the dissociative bis-allyl region of the Cg), cut are
much too high. The differences between the CCSD energies at R = 4.0 A and at the
R values that correspond to minima on the Cg), PES cuts are 45.5 and 48.0 kcal/mol
when the 6-31G* and 6-311G** basis sets are employed. This should be compared
to the analogous differences calculated from the CR-CCSD(T) and MCQDPT data,
of 35.7 and 26.4 kcal/mol, respectively, for the 6—31G* basis set, and 39.3 and 27.5
kcal/mol, respectively, for the 6-311G** basis set. This shows that in order to obtain
a correct and quantitative description Of the 03,, cut of the Cope rearrangement
of 1,5-hexadiene, one must include the effect of triple excitations in CC calculations.
However, including the triples through the standard CCSD(T) method fails, as can be
seen in Figure 28. The CCSD(T) method not only gives an incorrect description of the
transition state, as discussed above. It also fails in the dissociative bis-allyl region,
producing an unphysical hump on the PES. The CR-CCSD(T) approach removes
these failures by sealing down the large triples corrections in the diradical regions
with the denominator DIT).

Since the value of the denominator Dm is approximately 1.4 over a large range
of R distances, varying by 0.03 or less for between R = 1.7 and 2.5 A, it may seem
possible to obtain a correct description of the transition state region Of the PES
using a fixed value of Dm, such as DIT) = 1.4. The results of the CR-CCSD(T)
calculations with the denominator artificially fixed at DIT) = 1.0 and Dm = 1.4,
along with the results of the CCSD(T), CR-CCSD(T), and MCQDPT calculations,

are shown in Figure 28. The CR-CCSD(T)/D(T) = 1.4 curves shown in Figure 28 are

70

almost identical to the CR-CCSD(T) curve up to R z 2.5 A. However for R > 2.5
A the CR—CCSD(T)/D(T) = 1.4 and CR—CCSD(T) curves begin to diverge, with
the CR—CCSD(T)/D(T) = 1.4 method beginning to produce an unphysical hump in
the bis-allyl region of the surface which does not appear on the CR—CCSD(T) or
MCQDPT curves. The minimum on the CR-CCSD(T)/D(T) = 1.4 curve is very
near the minimum Of the true CR-CCSD(T) curve, and the corresponding activation
energies are nearly identical, but the overall description of the Cg), PES cut by the
CR-CCSD(T)/D(T) = 1.4 approach is incorrect. When the denominator DIT) is
fixed at a value of 1, the corresponding C2,, cuts are nearly identical to the curves
resulting from the standard CCSD(T) calculations, which have already been shown to
be incorrect. This shows that in order to obtain a balanced description of the entire
PES when using CC methods with singles, doubles, and non-iterative triples, the
triples energy correction must be sealed in a geometry-dependent manner, as is done
in CR—CCSD(T). It is always possible to contemplate empirical ways Of rescaling the
triples correction of the CCSD(T) approach in different regions of the PES to provide
improved results for the surface. The CR-CCSD(T) method provides a rigorous ab
initio recipe for doing this by relating the denominator Dm in the CR—CCSD(T)
triples correction to the singly and doubly excited cluster amplitudes Obtained from
the CCSD calculations. Since cluster amplitudes vary with geometry, and increase or
decrease in magnitude based on the amount of diradical character of the molecule, the
denominator DIT) Of the CR—CCSD(T) method produces the desired improvements
in the CCSD(T) results.

Although further analysis of the above observations may be necessary, the cor-

71

relation between the cluster amplitudes, which define the diradical character of a
molecule, and the denominator D‘T) used in the CR-CCSD(T) calculations, and the
changes in the value of DIT) relative to reactants, may prove to be a useful tool
when examining reaction pathways involving diradicals. A comparison of the values
of the denominator DIT) Obtained here and the effective numbers of unpaired elec-
trons reported in Ref. 138, which have been used to argue the aromatic character
of the transition state for the Cope rearrangement of 1,5-hexadiene state shows that
both quantities display similar patterns. As shown in Ref. 138 and its supporting
information, the values of the effective unpaired electrons Obtained from MRCI cal-
culations employing the 6-31G* basis set decrease from 2.85 at R = 1.54 A to 1.55
in the R z 2.0 A region, to increase again to 3.26 at R = 4.0 A. The values of the
denominator D“) from our CR—CCSD(T) calculations, using the same 6-31G* basis
set, show the same pattern. D‘T) decreases from 1.49 at R = 1.55 A to 1.37 at R
= 2.0 A, to increase again to 1.78 at R = 4.0 A. Thus, there seems to be a similarity
in the behavior of the denominator Dm and the number of effectively unpaired elec-
trons which can be used to determine the amount of diradical character in a given

molecular system.

6.4. Results for the Bergman Cyclization of Enediynes

The simplest enediyne structure is the C6H4 structure, labeled as System 1 in
Figure 25. As mentioned earlier, the DFT calculations employing the (U)B3LYP

functional and the 6-31G** basis set provide relatively stable results for the TS struc-

72

ture as well as the diradical product molecule. The results of (U)B3LYP, CCSD(T),
CR-CCSD(T), CCSD(TQf), and CR—CCSD(TQ) calculations using the 6-31G** basis
can be found in Table 12. The (U)B3LYP calculations produce a TS barrier energy,
AEI, of 31.2 kcal/mol above the energy of the reactant structure and an energy of 3.3
kcal/mol above the reactant molecule for the diradical Bergman cyclization product
molecule, AER. These values are reasonably close to the experimental values Of 30.1
:I: 0.5 kcal/mol and 7.8 :l: 0.7 kcal/mol, respectively. The CCSD method results in
reaction energetics which are well above the experimental energetics, producing a bar-
rier height of 37.2 kcal/mol and a diradical product energy of 26.5 kcal/mol above the
energy of the reactant, as seen in Table 12. This is not a surprise, since CCSD always
produces energies which are too high for regions of PESs where unpaired electrons
are separated by a considerable distance, as is the case in diradicals. For the reac-
tion of System 1, the standard CCSD(T) method produces energies which seem close
to experimental values. The AEI value for System 1, resulting from the CCSD(T)
approach is 29.3 kcal/mol, while the value of AER for system 1, as calculated with
the CCSD(T) approach, is 5.0 kcal/mol. The CR-CCSD(T) value of AE1 of 33.0
kcal/mol agrees with experiment rather well. However, the CR—CCSD(T) result for
the diradical product molecule of System 1 is too high when compared to experi-
ment. The CR—CCSD(T) calculations result in a diradical product energy AER Of
16.6 kcal/mol, when the experimental value is 7 .8 kcal/mol. When the corrections
due to quadruples are added in the form of the CCSD(TQf) and CR—CCSD(TQ)
methods, the results do not change very much when compared to the CCSD(T) and
CR-CCSD(T) results (see Table 12).

73

The results for System 2, the C5NH3 molecule, shown in Figure 25, follow the same
pattern as the results for System 1, as can be seen in Table 13. The (U)B3LYP values
of AE1 and AER for System 2 are 21.8 and -9.3 kcal/mol, respectively. The accuracy
of these (U)B3LYP values is not well known due to a lack of relevant experimental
data for the reaction of System 2. However, the DFT calculations with the (U)B3LYP
functional have been shown to provide results of reasonable quality in other reactions
Of this type,174 so we will use them for comparison purposes. Once again, the CCSD
method produces activation barrier and product energies, relative to the reactant,
which are too high (27.5 and 5.1 kcal/mol, respectively). The standard CCSD(T)
method produces TS and product energies which are close to those obtained with
(U)B3LYP (20.5 and -11.9 kcal/mol, respectively). The results obtained from the
CR—CCSD(T) approach lie slightly above the (U)B3LYP results, with AE1 and AER
energies of 23.8 and -2.4 kcal/mol, respectively, as shown in Table 13. Once again, as
in the case of System 1, adding the approximate corrections due to quadruply excited
clusters in the form Of the CCSD(TQf) and CR-CCSD(TQ) methods does not change
the results too much.

One might think that since the standard CCSD(T) method seems to provide
relatively good results for the Bergman cyclization Of Systems 1 and 2, it should also
provide the same quality Of the results for other Bergman cyclization reactions shown
in Figure 25. As can be seen from the results for Systems 3—7 in Tables 14—18, this is
not the case. For the ONE, molecule, System 3, the CCSD(T) approach produces
a value of AE1 which is significantly lower than that of the (U)B3LYP method. We

Obtain 22.8 kcal/mol in the CCSD(T) case and 26.2 kcal/mol in the (U)B3LYP case.

74

 

The situation becomes even more interesting when the AER values are examined. The
reaction energy, AER, calculated with the CCSD(T) method is -35.6 kcal/mol. This
is drastically lower than the (U)B3LYP value for the reaction energy of -5.8 kcal/mol,
a difference of almost 30 kcal/mol. The AE1 and AER values resulting from the CR-
CCSD(T) calculations seem to be much better than those from the standard CCSD(T)
calculations. The CR—CCSD(T) calculations produce 26.5 and 5.5 kcal/mol for AE1
and AER, respectively, in much better agreement with (U)B3LYP. When the effects
due to quadruply excited clusters are included by way of the CCSD(TQf) approach,
the CCSD(T) result for AEir changes very little. However, the CCSD(T) value of
AER increases by 6.3 kcal/mol, when the CCSD(TQf) quadruples correction is added
to the CCSD(T) energy. This large change in AER when going from CCSD(T) to
CCSD(TQf), is a sign of a failure of standard CC methods based on (divergent)
MBPT expansions. When the effects of quadruples are added to the CR—CCSD(T)
calculations in the form of the CR—CCSD(TQ) approach, the AE1 and AER values
change very little, which is the desired behavior for diradical systems. Clearly, the
conventional, MBPT-based CCSD(TQf) approach cannot overcome the failures of
CCSD(T).

The results for Systems 4—7 show similar behavior to the results Obtained for
System 3. The CCSD(T) and CR—CCSD(T) results for the activation energy AEI are
relatively close to the (U)B3LYP values, while the results of the CCSD(T) calculations
for the reaction energy are much too low when compared to the energies resulting from
the (U)B3LYP calculations. For instance, the AE1 value for System 7, calculated

with the CCSD(T) method, is 27.2 kcal/mol. This is close to the value of 28.6

75

 

kcal/mol from the (U)B3LYP calculations. Likewise the CR—CCSD(T) activation
energy for System 7 Of 31.7 kcal/mol is close to the activation energy obtained with
(U)B3LYP. The situation changes when the AER values for System 7 are examined.
The CCSD(T) value Of -25.5 kcal/mol is almost 28 kcal/mol below the result Of the
(U)B3LYP calculations. The CR-CCSD(T) calculations place the product of the
cyclization reaction involving System 7 21.7 kcal/mol above the reactant molecule,
compared to 2.7 kcal/mol Obtained with (U)B3LYP and -25.5 kcal/mol Obtained
with CCSD(T). Even though the CR-CCSD(T) result is quite high compared to the
(U)B3LYP results, it retains the same sign. When the effects due to the quadruply
excited clusters are incorporated into the CC calculations for Systems 4—7, the results
for the AE1 values do not change much when compared with the CC with singles,
doubles, and triples, as can be seen in Tables 15—18. The situation changes when
AER values are examined. In this case, we observe rather significant changes, when
going from CCSD(T) to CCSD(TQf), which indicates problems with the standard
CCSD(T) and CCSD(TQf) methods, and almost no changes when going from CR-
CCSD(T) to CR-CCSD(TQ), which is a more desirable behavior when diradicals are
examined.

When looking at the results of the calculations for the reactions listed in Figure
25, shown in Tables 12—18, one might think that the CR—CCSD(T) approach does not
perform as well as in other cases. In reality, however, the CR-CCSD(T) results are
quite likely the most accurate for the basis set employed in the calculations. There are
a few reasons for this. First, for most systems shown in Figure 25, the only exception

being System 1, there are no experimental data to compare with. For this reason,

76

.' mi! 2:

 

we can only compare the CR-CCSD(T) results with the results of other theoretical
calculations, such as the DFT calculations. Whether the DFT calculations using the
(U)B3LYP approach are accurate or not is itself debatable. As recently shown by
Cramer et. al.,126 the hybrid functionals Of DFT, such as (U)B3LYP, may provide
highly undesirable results in calculations of relative energetics involving closed-shell
(in our case, reactants) and diradical (in our case intermediate) structures. Pure
DFT functionals may work better, but there is no guarantee that this is generally the
case. Another consideration is that the geometries of the structures used in our CC
calculations were optimized using (U)B3LYP. It is possible that these geometries could
be relatively far removed from the Optimum CR-CCSD(T) geometries. Unfortunately,
it was not feasible to Optimize the geometries at the CC levels Of theory due to the
large computer cost of finding so many stationary points on the relevant PESs. One
might, of course, think that there are some problems with CR-CCSD(T). For example,
the CR-CCSD(T) method is not size-extensive, which means that one begins to lose
accuracy as systems become very large. However, from our experience, the size-
extensivity errors in the CR-CCSD(T) calculations are on the order of 0.5—1% of
the correlation energy (changes in the correlation energy along the reaction path if
relative energies are examined)“:62 Small errors like this cannot have a significant
effect on the calculation for medium size Systems 1—7. It is also worth mentioning
that the CCSD(T) approach is size-extensive and it fails miserably in most of the
calculations for the diradical molecules shown here (see Tables 12—18). Since the
exact energies for these reactions are not known, the only solution at this time is to

continue the advancement of theoretical methods that can be used in calculations for

77

 

Systems 1-7. Piecuch et. al. have recently come up with rigorously size-extensive
CR-CC methods, such as CR-CC(2,3),°7’°8 which may help to resolve the issue of how
accurate our CR-CCSD(T) and CR-CCSD(TQ) results for the Bergman cyclizations
are. The CR-CC(2,3) approach is as accurate or, often, more accurate than the
CCSD(T) method for closed-shell molecules near the equilibrium, while breaking
bonds and describing diradicals as accurately as full CCSDT (i.e. almost exactly).
The CR-CC(2,3) method is the most accurate non-iterative triples approach (more
accurate than CR-CCSD(T)). It will be interesting to recalculate the AE1 and AER

values with CR-CC(2,3) in the future.

6.5. Conclusion

In summary, the CR-CCSD(T) method can produce very good results for reactions
involving diradical structures. This has been shown for the both the Cope rearrange-
ment of 1,5-hexadiene and Bergman cyclizations of enediyne molecules. In the case of
the Cope rearrangement Of 1,5-hexadiene, the CR-CCSD(T) approach favors a con-
certed mechanism through an aromatic TS. This agrees with earlier multi-reference
perturbation theory calculations and experimental studies. This should be contrasted
with the standard CCSD(T) method, which predicts a stepwise pathway through a
1,4-diyl diradical intermediate. In the case of the Bergman cyclization Of enediynes,
the CR-CCSD(T) and CR-CCSD(TQ) methods are shown to remove the failing of the
standard CCSD(T) and CCSD(TQf) approaches for the diradical cyclization product

molecules, although further studies are required to assess the accuracy of the CR—CC

78

 

I ‘T‘JI

calculations. It has also been shown that the denominator DIT), which results from
the CR-CCSD(T) calculations, correlates with the degree of diradical character of the

system of interest.

 

79

7. Summary and Concluding Remarks

The CR-CCSD(T) and other CR-CC methods were developed to provide easy-to-
use “black-box” approaches which can provide excellent results in the bond breaking
and diradical regions of PESs. Normally one has to use expensive and often imprac-
tical multi-reference techniques to get adequate accuracy in these regions Of PESs,
since standard “black-box” approaches, such as CCSD(T) (not to mention lower-
Order methods), fail when bond breaking and diradicals are considered. In this work
the CR-CCSD(T) has been shown to remove the failing Of the standard CCSD(T)
approach and provide results which are very close to those Of the most advanced
multi-reference methods, such as MRCI(Q).

The CR—CCSD(T) method has been shown to produce results for the Be + HF
—+ BeF + H, BeH + F, and HBeF reactions which mimic the results of exact, full
CI and accurate MRCI(Q) calculations. In the regions of the BeHF PES where the
H—F distance is stretched, the standard CCSD(T) approach produces huge errors
relative to the full CI and MRCI(Q) results, whereas the CR—CCSD(T) results in
these regions, as well as all other regions, are practically identical to the full CI
and MRCI(Q) results. This means that one can Obtain highly accurate reactive
PESs without the hassle Of selecting active orbitals in a subjective manner, as in the
MRCI(Q) approach, and without the high costs of MRCI(Q) calculations.

The CR—CCSD(T) approach also provided excellent results regarding the vibra-
tional spectrum Of the partly diradical trans-HNOO system. In consequence, the CR-

CCSD(T) method could be used to resolve a controversy related to the discovery of

80

 

HNOO in matrix isolation studies, reported in Refs. 104 and 105. The CR-CCSD(T)
results for the vibrational spectrum, along with the results of other high quality CC
calculations, helped us show that the correct vibrational assignments were those of
Laursen, Grace, DeKock, and Spronk.1°5 The CR-CCSD(T) method performed very
well in this case, even though the trans—HNOO molecule has some diradical character,
a situation which is difficult to describe using single reference RHF-based methods.
Finally, the CR—CCSD(T) method has been used to calculate energies for the re-
actants, TSs, intermediates, and products of reactions which involve diradical struc-
tures, namely the Cope rearrangement of 1,5-hexadiene and the Bergman cyclization
Of enediynes. The Cope rearrangement Of 1,5-hexadiene has been the source Of con-
troversy, with different theoretical methods providing differing mechanistic pathways.
The CR-CCSD(T) calculations resulted in producing a pathway through an aromatic
TS which agrees with multi-reference perturbation theory calculations and experi-
mental data, whereas the standard CCSD(T) method produced an incorrect pathway
through a diradical structure. The value of the denominator Dm, calculated in the
CR—CCSD(T) method, was shown to correlate with the degree Of diradical character
of the system of interest. For the Bergman cyclization of enediynes, which have shown
promise as antitumor antibiotics, the CR-CCSD(T) and CR—CCSD(TQ) methods re-
move the failing of the standard CCSD(T) and CCSD(TQf) approaches in calculations
involving the cyclization product, where the molecule has a large amount of diradical
character, but further studies may be required to assess the accuracies of the CR-CC

calculations for these systems.

81

 

Appendices

82

 

Appendix A. The PESs for the BeF H system as described by the

MIDI basis set

Table 19. The CCSD, CCSD(T), R-CCSD(T), CR-CCSD(T), and full CI energies, in
hartree, of the collinear BeFH system, Obtained with the MIDI basis set. Distances

 

 

 

are in bohr.

RBe_p RF_H CCSD CCSD(T) R-CCSD(T) CR-CCSD(T) Full CI
1.8 1.2 -113.265170 —113.265394 -113.265388 -113.265374 -113.265580
1.8 1.4 -113.418745 -113.419047 -113.419039 -113.419020 -113.419259
1.8 1.6 -113.480214 -113.480629 -113.480617 -113.480591 -113.480869
1.8 1.7325 -113.493946 -113.494457 -113.494441 -113.494410 -113.494716
1.8 1.8 -113.495809 -113.496376 -113.496358 -113.496323 -113.496647
1.8 2.0 -113.488826 -113.489625 -113.489595 -113.489552 -113.489939
1.8 2.25 -113.469618 -113.471541 -113.471374 -113.471353 -113.471957
1.8 2.5 -113.462690 -113.472896 -113.469425 -113.468795 -113.469185
1.8 2.75 -113.469458 -113.484243 -113.477603 -113.476359 -113.476754
1.8 3.0 -113.476874 -113.489436 -113.483722 -113.482782 -113.483354
1.8 3.5 -113.486698 -113.494820 -113.491185 -113.490743 -113.491313
1.8 4.0 -113.491572 -113.497276 -113.494667 -113.494401 -113.494870
1.8 5.0 -113.494345 -113.497981 -113.496215 -113.496042 -113.496408
1.8 6.0 -113.494281 -113.497320 -113.495788 -113.495636 -113.495998
1.8 8.0 -113.494115 -113.497084 -113.495527 -113.495384 -113.495759
1.9 1.2 -113.397230 -113.397445 -113.397440 -113.397426 -113.397634
1.9 1.4 -113.551599 -113.551892 -113.551884 -113.551866 -113.552105
1.9 1.6 -113.613562 -113.613968 —113.613956 -113.613930 -113.6l4207
1.9 1.7325 -113.627508 -113.628009 -113.627994 -113.627963 -113.628267
1.9 1.8 -113.629467 -113.630024 -113.630006 -113.629972 -113.630293
1.9 2.0 —113.622833 -113.623632 -113.623601 -113.623559 -113.623944
1.9 2.25 -113.604545 -113.606526 -113.606349 -113.606323 -113.606926
1.9 2.5 -113.597827 -113.606954 -113.604060 -113.603507 -113.603980
1.9 2.75 -113.603783 —113.617225 -113.611482 -113.610357 -113.610805
1.9 3.0 -113.610730 -113.622603 -113.617371 -113.616465 -113.617047
1.9 3.5 -113.620132 -113.628104 -113.624594 -113.624142 -113.624705
1.9 4.0 -113.624835 -113.630540 -113.627959 -113.627684 -113.628152
1.9 5.0 -113.627556 -113.631280 -113.629483 -113.629305 -113.629680
1.9 6.0 -113.627515 -113.630667 -113.629084 -113.628927 -113.629300
1.9 8.0 -113.627354 -113.630449 -113.628831 -113.628682 -113.629066
2.0 1.2 -113.495505 -113.495715 -113.495710 -113.495696 -113.495906
2.0 1.4 -113.650493 —113.650783 -113.650775 -113.650756 -113.650997

 

 

83

 

W

Table 19. cont’d

 

 

 

1213,.F 123-3 CCSD CCSD(T) R-CCSD(T) CR—CCSD(T) Full CI

2.0 1.6 413.712377 413.713230 413.713263 413.713242 413.713520
2.0 1.7325 413727042 413.727541 413.727526 413727495 413.727300
2.0 1.3 413729111 -113.729668 413729650 413729616 413729937
2.0 2.0 413.722330 413723633 413723657 413723615 413724002
2.0 2.25 413705307 413707234 413707110 413707073 413707630
2.0 2.5 413.693052 413.706013 413703730 413703274 413703343
2.0 2.75 413702301413715003 413710110 413709105 413709622
2.0 3.0 413709100 413720419 413715626 413714743 413715353
2.0 3.5 413717930 413725905 413722466 413721992 413722563
2.0 4.0 413722400 413.723261 413725646 413725353 413725332
2.0 5.0 413725017 413723931 413727033 413726390 413727233
2.0 6.0 413724977 413.723395 413726637 413726516 413726907
2.0 3.0 413724313 413.723193 413726433 413726270 413726672
2.2 1.2 413.621163 413621372 413.621367 413.621353 413.621563
2.2 1.4 413777052 413777345 413777337 413777313 413777564
2.2 1.6 413340179 413.340591 413340579 413.340552 413.340334
2.2 1.7325 413354324 413355333 413355322 413355239 413.355599
2.2 1.3 413357154 413357729 413357710 413357675 413353002
2.2 2.0 413351324 413352665 413352633 413352533 413352934
2.2 2.25 413335194 413337069 413336923 413336377 413337473
2.2 2.5 413325726 413331616 413330326 413330047 413330304
2.2 2.75 413327109 413337043 413333693 413332935 413333631
2.2 3.0 413331514 413341901 413337961 413337146 413337365
2.2 3.5 413333652 413346930 413343593 413343049 413343702
2.2 4.0 413342443 413349040 413346203 413.345341 -113.846390
2.2 5.0 -113.844671-113.849595 413347235 413347032 413347492
2.2 6.0 413344562 413349027 413346321 413346593 413347049
2.2 3.0 413344366 413343391 413346547 413346327 413346790
2.4 1.2 413637409 413637623 413637613 413637603 413637324
2.4 1.4 413343377 413344130 413344172 413344152 413344403
2.4 1.6 413907634 413903064 413903052 413903023 413903312
2.4 1.7325 413922764 413923304 413923237 413923252 413923571
2.4 1.3 413925366 413925972 413925953 413925914 413926251
2.4 2.0 413920900 413921733 413921749 413921700 413922103
2.4 2.25 413904722 413906470 413906354 413906295 413906392
2.4 2.5 413392550 413396969 413396263 413396094 413396943
2.4 2.75 413339709 413397660 413395542 413395019 413395903
2.4 3.0 413391333 413900307 413397731 413397021 413397903

 

 

84

Table 19. cont’d

 

 

 

R334 RF.H ccso CCSD(T) R-CCSD(T) CR-CCSD(T) Full CI

2.4 3.5 -113.895868 413904303 413901459 413900337 413901634
2.4 4.0 413393600 413906339 413903220 413902743 413903427
2.4 5.0 -113.900131-113.906671 413903634 413903326 413903904
2.4 6.0 413399343 413906111 413903059 413902730 413903299
2.4 3.0 413399577 413906033 413902740 413902421 413902939
2.5 1.2 413707319 413707536 413707531 413707516 413707739
2.5 1.4 413363966 413364276 413364263 413364246 413364501
2.5 1.6 413927971413923412 413923399 413923370 413923662
2.5 1.7325 413943314 413.943363 413943351 413943314 413944137
2.5 1.3 413.946036 413946659 413946639 413946599 413946940
2.5 2.0 413941932 413942334 413942799 413942743 413943162
2.5 2.25 413925330 413927523 413927413 413927353 413927947
2.5 2.5 413912335 413.916223 413915703 413915556 413916426
2.5 2.75 413907261-113914352 413912705 413912230 413913251
2.5 3.0 413907199 413916170 413913513 413912377 413913351
2.5 3.5 413910026 413919263 413915931 413915331 413916223
2.5 4.0 413.912043 413920535 413917194 413916655 413.917432
2.5 5.0 413913035 413920650 413917250 413916323 413917436
2.5 6.0 413912659 413920103 413916513 413916116 413916765
2.5 3.0 413912342 413920173 413916164 413915773 413916422
2.5719 1.2 413713039 413713309 413713303 413713233 413713339
2.5719 1.4 413374321-113375135 413375126 413375105 -113.875361
2.5719 1.6 413933963 413.939417 413939404 413939373 413939669
2.5719 1.7325 413954442 413955007 413954939 413954952 413955273
2.5719 1.3 413957241413957375 413957354 413957313 413.953153
2.5719 2.0 413953355 413954270 413954235 413954132 413954600
2.5719 2.25 413937254 413933913 413933315 413933746 413939337
2.5719 2.5 413922352 413926437 413926004 413925370 413926741
2.5719 2.75 413916136 413922720 413921352 413920936 413922005
2.5719 3.0 413.914315 413923409 413921050 -113.920462 413921501
2.5719 3.5 413916249 413925635 413922463 413921300 413922763
2.5719 4.0 413917676 413926677 413923224 -113.922643 413923493
2.5719 5.0 413913233 -113.926661-113.922945 413922464 413923200
2.5719 6.0 413917742 413926142 413922116 413921664 413922333
2.5719 3.0 413917330 -113.926286 413921744 413921303 413922012
2.6 1.2 413721667 413721337 413721331 413721366 413722092
2.6 1.4 413373421—113373737 413373723 413373707 413373964
2.6 1.6 413942615 413943067 413943054 413943023 413943319

 

 

85

 

Table 19. cont’d

 

 

RB,_F RH, CCSD CCSD(T) R-CCSD(T) CR-CCSD(T) Full CI

2.6 1.7325 413953136 413953704 413953636 -113.958649 413953976
2.6 1.3 413960961-113961600 413961579 413961537 413961333
2.6 2.0 413957151413953071 413953035 413957932 413953401
2.6 2.25 413941040 413942635 413942590 413942519 413943110
2.6 2.5 413926293 413929772 413929369 413929233 413930103
2.6 2.75 413919016 413925346 413924074 413923723 413924763
2.6 3.0 413917116 413925560 413923312 413922744 413923309
2.6 3.5 413917970 413927473 413924231 413923615 413924615
2.6 4.0 413919143 413923345 413924353 413924256 413925144
2.6 5.0 413919573 413.923275 413924433 413923930 413924693
2.6 6.0 413913975 413927767 413923565 413923090 413923340
2.6 3.0 413.913594 413927947 413923133 413922720 413923453
2.7 1.2 413732149 413732371 413732366 413732350 413732573
2.7 1.4 413333942 413339264 413339255 413339233 413339493
2.7 1.6 413953263 413953726 413953712 413953630 413953930
2.7 1.7325 413.963921413969503 413969435 413969446 413969777
2.7 1.3 413971327 413972431 413972459 413972417 413972767
2.7 2.0 -113.968241-113.969176 413.969140 413969034 413.969503
2.7 2.25 413952046 413953652 413953563 413953433 413954075
2.7 2.5 413936135 413939233 413933970 413933346 413939705
2.7 2.75 -113.926751-113.932418 413931435 413.931149 413932225
2.7 3.0 413922925 413930322 413923942 413923447 413929596
2.7 3.5 413921554 413931233 413923134 413927517 413923634
2.7 4.0 413921732 413931625 413923013 413927369 413923336
2.7 5.0 413921432 413931344 413927053 413926469 413927364
2.7 6.0 413920625 413930392 413926033 413925477 413926350
2.7 3.0 413920166 413931217 413925621 413925072 413925926
2.9 1.2 413.746131413746357 413746351 413746335 413746565
2.9 1.4 413902330 413903160 413903151 413903123 413903391
2.9 1.6 413967223 413967701 -113.967687 413967654 413967959
2.9 1.7325 413933030 -113.983634 413933615 413933574 413.933913
2.9 1.3 -113.986031-113.986710 413936633 413936643 413937001
2.9 2.0 413932635 413933644 413933607 413933543 413933930
2.9 2.25 413966152 413967695 413967616 413967533 413963113
2.9 2.5 413943179 413950374 413.950655 413950533 413951357
2.9 2.75 413935030 413939656 413939045 413933323 413939927
2.9 3.0 413927343 413934155 413932373 413932499 413933773
2.9 3.5 413920950 413930690 413923020 413927393 413.923755

 

 

 

 

 

86

Table 19. cont’d

 

 

 

133..-,» RH, ccso CCSD(T) R-CCSD(T) CR—CCSD(T) 1151101

2.9 4.0 413913705 413929752 413926070 413925354 413926671
2.9 5.0 -113.916561-113.928945 413923312 413923067 413924235
2.9 6.0 413915217 413923673 413922467 413.921720 413922915
2.9 3.0 -113.914561-113.929393 413921935 413921239 413922404
3.1 1.2 413755403 413755623 413755622 413755607 413755337
3.1 1.4 413911341413912175 413912166 413912143 413912403
3.1 1.6 413976112 413976600 413976535 413976551 -113.976859
3.1 1.7325 413991925 413992543 413.992523 413.992431 413992324
3.1 1.3 413994944 413.995633 413995615 413995569 413995932
3.1 2.0 413991603 413992530 413.992543 413992431 413992913
3.1 2.25 413974534 413976031 413976003 413975919 413976493
3.1 2.5 413954339 413957297 413957123 413957000 413957733
3.1 2.75 413933669 413942572 413942157 413941963 413943030
3.1 3.0 413927431-113933315 413932430 413932127 413933450
3.1 3.5 413915744 413925123 413922904 413922346 413923933
3.1 4.0 -113.910601-113.922329 413913730 413913043 413919692
3.1 5.0 413906130 413920730 413914933 413914093 413915721
3.1 6.0 413904104 413920793 413913296 413912351 413913967
3.1 3.0 413903133 413922039 413912763 413911794 413913372
3.3 1.2 413762729 413762952 413762946 413762931 413763161
3.3 1.4 413913323 413919156 413919147 413919123 413919339
3.3 1.6 413932346 413933339 413933324 413933290 413933599
3.3 1.7325 413993565 413999190 413999170 413999127 413999472
3.3 1.3 414.001543 414.002250 414.002227 414.002179 414.002545
3.3 2.0 413993066 413999043 413999006 413993941 413999332
3.3 2.25 413930432 413931946 413931376 413931733 413932353
3.3 2.5 413959367 413961595 413961450 413961317 413962077
3.3 2.75 413940737 413944162 413943345 413943652 413944673
3.3 3.0 -113.926633 413931697 413931046 413930767 413932073
3.3 3.5 413909349 413913597 413916737 413916230 413913023
3.3 4.0 413901613 413913506 413910222 413909493 413911444
3.3 5.0 413394503 413910972 413904739 413903715 413905779
3.3 6.0 413391700 413911366 413902734 413901567 413903653
3.3 3.0 413390466 413913266 413902200 413900944 413902997
3.5 1.2 413769150 413769369 413769363 413769343 413769573
3.5 1.4 413924377 413925203 413925199 413925175 413925441
3.5 1.6 413933597 413939039 413939074 413939039 413939350
3.5 1.7325 -114.004165 414004791 414004771 414004727 414005073

 

 

87

 

Table 19. cont’d

 

 

 

 

R534 123-3 CCSD CCSD(T) R-CCSD(T) CR-CCSD(T) 551101

3.5 1.3 414007079 414.007732 414007759 414007711 414003073
3.5 2.0 414003365 414.004341 414004304 414004233 414.004679
3.5 2.25 413935213 413936656 413936537 413936492 413937053
3.5 2.5 413962943 413965061 413964923 413964791 413965533
3.5 2.75 413942470 413945537 413945320 413945123 413946103
3.5 3.0 413926034 413930595 413930072 413929792 413931063
3.5 3.5 413904999 413.913055 413911543 413911043 413.912372
3.5 4.0 413393357 413905536 413902524 413901770 413903957
3.5 5.0 413334147 413901901 413395363 413394172 413396645
3.5 6.0 413330563 413902731 413393145 413391630 413394226
3.5 3.0 413.373994 413905475 413392676 413391019 413393525
3.7 1.2 413774949 413775163 413775153 413775143 413775372
3.7 1.4 413930327 413930653 413930645 413930621 413930337
3.7 1.6 413993734 413994222 413994203 413994173 413994433
3.7 1.7325 414009129 414009752 414009732 414009639 414010035
3.7 1.3 414011962 414.012662 414012639 414012590 414012957
3.7 2.0 414007933 414003954 414.003917 414003350 414009291
3.7 2.25 413939313 413990729 413990662 413990565 413991123
3.7 2.5 413966117 413963157 413963030 413967390 413963619
3.7 2.75 413944245 413947171 413946923 413946725 413947676
3.7 3.0 413.926142 413930233 413929330 413929540 413930769
3.7 3.5 413901639 413909149 413907319 413907290 413909133
3.7 4.0 413333145 413399450 413396627 413395796 413393123
3.7 5.0 413376267 413394332 413333052 413336595 413339373
3.7 6.0 413372035 413396417 413335773 413333370 -113.886765
3.7 3.0 413370227 413900061 413335463 413333213 413336056
3.9 1.2 413730145 413730355 413730349 413730335 413730564
3.9 1.4 413935212 413935535 413935526 413935503 413935763
3.9 1.6 413993326 413993310 413993796 413993761 413999071
3.9 1.7325 414013550 414014169 414014149 414014105 414014451
3.9 1.3 414016300 414016995 414016973 414016923 414017290
3.9 2.0 414012060 414013023 414012936 414012919 414013359
3.9 2.25 413992930 413994329 413994262 413994165 413994724
3.9 2.5 413969006 413970997 413970374 413.970731 413971451
3.9 2.75 413946095 413943903 413943672 413.943465 413949396
3.9 3.0 413926735 413930651 413930225 413929926 413931122
3.9 3.5 413399799 413906323 413905535 413905017 413906336
3.9 4.0 413334434 413395432 413392692 413391751 413394123

 

 

 

88

Table 19. cont’d

 

 

 

123-3 RH, CCSD CCSD(T) R-CCSD(T) CR—CCSD(T) 551101

3.9 5.0 413371120 413390346 413333130 413331323 413334233
3.9 6.0 413366620 413392735 413330931 413373502 413331539
3.9 3.0 -113.864611-113.897480 413330927 413377337 413330393
4.1 1.2 413734709 413734916 413734911 413734396 413735125
4.1 1.4 413939512 413939331 413939323 413939300 413940064
4.1 1.6 414002366 414002346 414002332 414002793 414003107
4.1 1.7325 414017434 414013043 414.013029 414017935 414.013330
4.1 1.3 414020107 414020793 414020775 414020726 414021091
4.1 2.0 414015629 414.016535 414016549 414016431 414016919
4.1 2.25 413996112 413997497 413997431 413997332 413997333
4.1 2.5 413971622 413973531 413973459 413973315 413974029
4.1 2.75 413947944 413950631 413950456 413950245 413951163
4.1 3.0 413927679 413931453 413931045 413930740 413931911
4.1 3.5 413393935 413905725 413904524 413903923 413905704
4.1 4.0 413332470 413393159 413390420 413339376 413391737
4.1 5.0 413363234 413337933 413330316 413373154 413331149
4.1 6.0 413363737 413391352 413373441 413375337 413373460
4.1 3.0 413361729 413397265 413373705 413374736 413377304
4.5 1.2 413791969 413792171 413792167 413792152 413792330
4.5 1.4 413946375 -113.946689 413946630 413946653 413946922
4.5 1.6 414003323 414009297 414009233 414009243 414.009557
4.5 1.7325 414023639 414024246 414024226 414024133 414024527
4.5 1.3 414026137 414026370 414026347 414026793 414027163
4.5 2.0 414.021331414022275 414022239 414022171 414022607
4.5 2.25 414001243 414002607 414002542 414.002441 414002993
4.5 2.5 413975933 413977909 413977733 413977642 413973347
4.5 2.75 413951334 413954001 413953779 413953564 413954465
4.5 3.0 413929960 413933614 413933203 413932395 413934036
4.5 3.5 413399205 413905632 413904434 413903351 413905553
4.5 4.0 413331439 413391330 413.339051 413337335 413390136
4.5 5.0 413366324 413337197 413373697 413376006 413373346
4.5 6.0 413362601413392309 413377475 413373359 413376273
4.5 3.0 413360347 413900232 413373361 413372921 -113.875695
4.7 1.2 413794752 413794953 413794943 413.794934 -113.795162
4.7 1.4 413949014 413949326 413949313 413949295 413949559
4.7 1.6 414011311414011732 414011769 414011734 414012043
4.7 1.7325 414026032 414026635 414026617 414026573 414026916
4.7 1.3 414023533 414029212 414029190 414029141 414029505

 

 

 

89

‘ ‘n..' 1‘ I

"l.

Table 19. cont’d

 

 

 

 

523,.F RH CCSD CCSD(T) R-CCSD(T) CR-CCSD(T) 551101

4.7 2.0 414023535 414024475 414024439 414.024371 414024306
4.7 2.25 414003246 414004603 414004537 414004437 414004937
4.7 2.5 413977739 413979649 413979529 413979331 413930034
4.7 2.75 413952732 413955433 413955210 413954994 413955391
4.7 3.0 413931073 413934706 413934299 413933935 413935116
4.7 3.5 413399739 413906170 413904962 413904325 413906000
4.7 4.0 413331671 413392071 413339193 413333003 413390194
4.7 5.0 413367103 413337664 413373334 413376054 413373774
4.7 6.0 413.363033 413393333 413377932 413373499 413.376273
4.7 3.0 413361492 413901997 413379059 413373095 413375713
5.0 1.2 413793042 413793242 413793237 413793223 413793451
5.0 1.4 413.952141413952451 413952443 413952421 413952635
5.0 1.6 414014264 414014733 414014720 414014635 414014993
5.0 1.7325 414023374 414.029475 414029456 414029412 414029756
5.0 1.3 414031320 414031996 414031974 414031925 414032239
5.0 2.0 414026160 414027096 414027060 414026991 414027426
5.0 2.25 414005647 414006997 414006932 414006331 414007330
5.0 2.5 413979373 413931772 413931652 413931504 413932204
5.0 2.75 413954604 413957240 413957017 413956300 413957691
5.0 3.0 413932567 413936173 413935769 413935453 413936574
5.0 3.5 413900660 413907070 413905346 413905210 413906347
5.0 4.0 413332326 413392714 413339731 413333595 413390692
5.0 5.0 -113.867901-113.888591 413379564 413376643 413379191
5.0 6.0 413364160 413394374 413373903 413374207 413376770
5.0 3.0 413362772 413904133 413330236 413.373330 413376246
5.2 1.2 413799740 413799939 413799934 413799920 413300149
5.2 1.4 413953757 413954067 413954059 413954036 413954300
5.2 1.6 414015792 414016260 414016247 414016212 414016521
5.2 1.7325 414030346 414030946 414030927 414030333 414031227
5.2 1.3 414032765 414033440 414033417 414033363 414033732
5.2 2.0 414027523 414023457 414023421 414023353 414023737
5.2 2.25 414006902 414003249 414003134 414003033 414003631
5.2 2.5 413931002 413932397 413932777 413932623 413933327
5.2 2.75 413955591—113953222 413957999 413957731 413953671
5.2 3.0 413933406 413937012 413936601 413936235 413937402
5.2 3.5 -113.901251-113.907663 413906431 413905796 413907413
5.2 4.0 413332310 413393207 413390243 413339071 413391117
5.2 5.0 413363473 413339173 413330030 -113.877160 413379595

 

 

 

90

Table 19. cont’d

 

 

1113,.F 521.4,, CCSD CCSD(T) R-CCSD(T) CR—CCSD(T) 551101

5.2 6.0 413364332 413395645 413.379542 413374773 413377215
5.2 3.0 413363607 413905120 413331034 413374403 413376705
5.5 1.2 413301637 413301335 413.301330 413301366 413302095
5.5 1.4 413.955611413955920 413955912 413955390 413956154
5.5 1.6 414017546 414013014 414013001 414017966 414013275
5.5 1.7325 414032033 414032636 414032617 414032574 414032917
5.5 1.3 414034425 414035093 414.035076 414035027 414035391
5.5 2.0 414029094 414.030025 414029990 414029921 414.030355
5.5 2.25 414003355 414009699 414009634 414009532 414010030
5.5 2.5 413932322 413934213 413934093 413933944 413934642
5.5 2.75 413956764 413959391 413959167 413953943 413959336
5.5 3.0 413934427 413933031 413937613 413937301 413933414
5.5 3.5 413902022 413903447 413907203 413906571 413903165
5.5 4.0 413333475 413393396 413390913 413339751 413391734
5.5 5.0 413369255 413339392 413330736 413377911 413330202
5.5 6.0 413365324 413396421 413330352 413375597 413377364
5.5 3.0 413364632 413905975 413331932 413375232 413377370
6.0 1.2 413303725 413303922 413303913 413303903 413304132
6.0 1.4 413957552 413957360 413957352 413957329 413953094
6.0 1.6 414019333 414019349 414019335 414019301 414020110
6.0 1.7325 414033303 414034405 414034336 414034342 414034636
6.0 1.3 414036163 414036335 414036313 414036764 414037123
6.0 2.0 414.030741414031671 414031635 414031566 414032001
6.0 2.25 414009339 414011230 414011165 414011063 414011612
6.0 2.5 413933732 413935620 413935500 413935350 413936043
6.0 2.75 413953039 413.960663 413960433 413960219 413961106
6.0 3.0 413935565 413939170 413933755 413933436 -113.939546
6.0 3.5 413902933 413909335 413903130 413907493 413909071
6.0 4.0 413334293 413394761 413391759 413390616 413392526
6.0 5.0 -113.870191-113.890608 413331646 413373395 413330994
6.0 6.0 413366930 413396345 413331231 413376665 413373697
6.0 3.0 413.365931413906062 413332311 413.376302 413373217
3.0 1.2 413305423 413305616 413305611 413305597 413304629
3.0 1.4 413959131 413959433 413959425 413959403 413959667
3.0 1.6 414020343 414021304 414021291 414021257 414021565
3.0 1.7325 414035200 414035792 414035773 414035730 414036073
3.0 1.3 414037524 414033191 414033169 414033120 414033434
3.0 2.0 414032023 414032943 414032913 414032344 414033279

 

 

91

a!

Table 19. cont’d

 

 

 

123-5 RH, CCSD CCSD(T) R—CCSD(T) CR-CCSD(T) 551101

3.0 2.25 414011033 414012424 414012360 414012257 414012307
3.0 2.5 413934350 413936735 413936615 413936464 413937164
3.0 2.75 413959075 413961693 413961472 413961251 413962140
3.0 3.0 413.936521413940129 413939711 413939391 413940501
3.0 3.5 413903775 413910249 413903930 413903343 413909903
3.0 4.0 413335107 413395614 413392593 413391473 413393309
3.0 5.0 413371137 413390913 413332431 413330000 413331309
3.0 6.0 413363053 413395103 413331732 413377992 413379550
3.0 3.0 413367217 413900797 413332650 413377743 413379031

 

 

92

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97

Table 6. Energies (E) and geometries (R394: and R347) of the saddle
points on the BeFH PES for the Be—F—H angles 6 = 45°, 70°, 80°, 90°,
135°, and 180°, and energies (E) and geometries (R394 and R3941) of
the HBeF insertion minimum resulting from the CCSD(T), CR-CCSD(T),
and MRCI(Q) calculations with the cc-pVTZ basis set. Energies are in

eV, relative to the Be + HF asymptote, and internuclear separations are in
bohn

 

 

 

6’ Quantity CCSD(T) CR-CCSD(T) MRCI(Q)
45° E 1.36 1.41 1.30
R394: 3.51 3.50 3.52
RH_F 2.35 2.35 2.34
700 E 0.61 0.64 0.58
R384: 2.98 2.96 3.01
RH_F 2.05 2.06 2.03
80° E 0.60 0.65 0.57
R39_p 2.78 2.77 2.79
RH_F 2.20 2.22 2.18
90° E 0.83 0.79 0.71
R394: 2.72 2.71 2.72
R34: 2.31 2.34 2.30
135° E 1.19 1.29 1.22
R394: 2.76 2.74 2.76
R34: 2.31 2.36 2.30
180° E 1.30 1.40 1.35
R394: 2.80 2.79 2.81
Rn_p 2.29 2.34 2.28
0° (HBeF minimum) E -3.92 -3.93 -3.98
R3e_p 2.59 2.58 2.59
R394; 2.49 2.49 2.49

 

 

98

Table 7. Experimental results for the vibra-
tional frequencies of trans-HNOO, in cm‘l, re-
ported in Refs. 104 and 105.

 

 

frequency LBSW LGDS
V1 (NH str) 3287.7 3165.5
V2 (HNO bend) not obs. 1485.5
V3 (NO str) 1381.6 1092.3
u. (00 str) 843.2 1054.5
115 (N00 bend) 670.1 not obs.
Vs (torsion) 790.7 764.0

 

 

99

Table 8. Optimized equilibrium geometries of trans-HNOO, resulting from the
force field analysis discussed in Section 5.2. Distances are in A and angles are

 

 

 

in degrees.

MGthOd Ro—o RN—O RN—H 9N—o—o 9H—N—o
CCSD(T) 1.2951 1.2993 1.0291 115.93 99.76

CR—CCSD(T) 1.2808 1.2880 1.0269 116.54 100.12
CCSD(TQf) 1.2901 1.3025 1.0284 115.97 99.63

CCSDT-3(Qf) 1.2859 1.3060 1.0286 115.99 99.50

 

 

100

 

 

 

 

 

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102

Table 11. Activation energies, AE*, and interallylic distances of the transition
states, R1, for the Cope rearrangement of 1,5-hexadiene.“

 

 

 

 

 

6-31G* 6-311G**

Method R1 (A) AEI (kcal/mol) R1 (A) AEI (kcal/mol)
CCSD 1.87 42.19 1.89 41.07
CCSD(T) 1.72 36.24 1.72 35.27

1.82 35.24
CR—CCSD(T) 1.83 38.91 1.86 37.73
CR-CCSD(T)/D<T> = 10" 1.73 37.13 1.72 36.23

1.83 36.07
CR-CCSD(T)/D<T> = 1.46 1.81 38.79 1.85 37.54
MCQDPT 1.86 30.95 1.88 28.31
Experimentd AEI = 35.0 kcal/mol

 

 

 

“ R1 is defined as the value of the interallylic distance R corresponding to
the minimum on the 02}; PES cut (note that each of the CCSD(T) and CR-
CCSD(T)/ D”) = 1.0 curves obtained with the 6-311G** basis set has two min-
ima). AEI is the energy at R = R1 relative to the reactant molecule. b The CR-
CCSD(T) approach in which the true, geometry-dependent, denominator D‘T)
in the (JR-CCSD(T) energy formula is replaced by a fixed value of Dm = 1.0.
c The CR-CCSD(T) approach in which the true, geometry-dependent, denomi-
nator Dm in the CR—CCSD(T) energy formula is replaced by a fixed value of
Dm = 1.4. d The experimentally derived result reported in Ref. 127.

103

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110

    
    
 

 

MI=

 

   

     
  

E-

A .
Lu
4:: i“
O -
D a-
v -.

k" '6’

m .
9‘ ... g==w=

 

 

 

 

 

 

 

 

 

R I” (bohr)

 

 

 

1.2 ‘

3 4 5 6
RMF (bohr)

Figure 1. Contour plots of the ground-state PES for the BeFH system resulting from
the (a) CCSD(T), (b) CR-CCSD(T), and (0) full CI calculations with the MIDI basis
set. The energies E are reported as E + 113.000 hartree. The thick contour line
representing E = —113.930 hartree separates the region where the contour spacing
is 0.010 hartree from the region where the contour spacing is 0.005 hartree. The
thick contour line representing E = —113.800 hartree separates the region where the
contour spacing is 0.005 hartree from the region where the contour spacing is 0.100
hartree. There is a contour line corresponding to E = ~113.931 hartree added to the
CCSD(T) PES to emphasize the presence of an artificial well, which does not appear
on the CR—CCSD(T) and full CI PESs.

Energy (hartree)

 

—0.02~f

Energy (hartree)
I

 

      
 

 

 

 

 

Mi

4 3
1:3:1‘030‘“)

Figure 2. The dependence of the differences between the (a) CCSD(T) and full CI
energies, and (b) CR—CCSD(T) and full CI energies for the BeFH system, as described
by the MIDI basis set, on the H—F and Be—F internuclear separations.

112

41392 ;~
73 - _
a: 113.93
‘-
1:
111
5
>
a: -113.94 ~
h
o
C
UJ
1
-113.95 ~
-113.96 ‘

vvtvfi—r

 

 

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1 .........
1 /,\,.—-3:—.-3-—-a
A\ I 1 1/1'
1539.5.-.Aerr 4* "A -113.89 ~1 ”9» 9 1:1
é U‘B- a B U ‘0‘ i ‘8“ IE]
1 :31 1
o _ -
E 113.93 1 11
co '1 I
R80_F=2.5719bohr 5 . 111 ‘1 “39-550 bohr
>3 1 '
accsom 2“ -113.97~ 1 1’ occsom
Ace-005mg 1 1 ACE-CCSD(T)
~FullCl uJ -- 1 1 Full CI
* -114.o1 — '1 f
(a) _ 11, (b)
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-114.01 1 1 1!
(C)
.. “111
-11405 AAAAAAAAAAAAAAAAAAAAAAAAA

 

 

1.0 20 3.0 4.0 5.0 6.0 7.0 3.0
R,” (bohr)

Figure 3. The potential energy curves of the collinear BeFH system along the H—F
stretch coordinate R114: at the Be—F distance RBe_ p fixed at (a) 2. 5719, (b) 5.0, and

(c) 8.0 bohr calculated with the (El ) CCSD(T ), (A ) CR—CCSD(T ), and (dotted line)
full CI approaches, and the MIDI basis set.

113

-1 13.60

 

 

 

 

 

 

 

“1* WWWW— 413.75 1
‘ T (a) . 1 (b)
“370 _ 1 accsom 1 accsom
' - 1 ACE-CCSD(T) 11330 , 1 Acn—ccsom
1 Full CI A ' - 1 Full CI
3 1 a ‘ ‘1
t 413.80- 1 t- 9 1
(U 1 g 1
5 11 ; 413.351 1 RH_,=5.0 bohr
> 1
9 _113.90 > V1I RH-F=1'7325 bOhf % 4 l1
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LU LU l Aétlammz—fi~-A~E-———Q
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414.00- ~ , A111
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414.10 1 1 - 1 1 1 413.95 1 . 1 , . .
2.0 3.0 4.0 5.0 6.0 7.0 3.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
R334: (bohr) RB” (bohr)
413.75 1
1
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1 mccsom
ACR-CCSD(T)
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9 1
t .
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9 1
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413.90 ~ 1 15153833311513 {31:91
%111
413.95

2.0 3.07.0 5.0 6.0 7.0 8.0

 

F1 B” (bohr)

Figure 4. The potential energy curves of the collinear BeFH system along the Be—F
stretch coordinate R3941 at the H—F distance R114: fixed at (a) 1.7325, (b) 5.0, and

(c) 8.0 bohr calculated with the (1:1) CCSD(T), (A) CR—CCSD(T), and (dotted line)
full CI approaches, and the MIDI basis set.

114

R E” (bohr)

   

2 3 4 5 6 7 8
RMF (bohr) RMF (bohr)

(C)

R I” (bohr)

 

2 3 4 5 6 7 8
R%F(bohr)

Figure 5. Contour plots of the ground-state PES for the BeFH system, at 0 = 180°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (RBe_p = 50.0 bohr and R34: = 1.7325 bohr). The thick contour line at 1.3
eV separates the region where the contour spacing is 0.3 eV from the region where
the contour spacing is 0.5 eV. There is a contour line corresponding to 0.12 eV added
to the MRCI(Q) PES to emphasize the depth of the product channel.

115

0.6
0.2
—0.2
—0.6
— l .0
—1.4

Energy (eV)

—l.8

 

—2.2

 

—2.6

 

Energy (eV)

    

 

 

5 6 5 4 3 --
”~F(b0br)7 8 8 7 {new (bob?)

Figure 6. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 180° BeFH system,
as described by the cc—pVTZ basis set, on the H~F and Be—F internuclear separations.

116

R ,H (bohr)

 

23456782345678
RMF (bohr) RBH. (bohr)

(C)

R H—F (bOhr)

 

2 3 4 5 6 7 8
RBH.(bohr)

Figure 7. Contour plots of the ground-state PES for the BeFH system, at 0 = 135°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc—pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (R384: = 50.0 bohr and RH_p = 1.7325 bohr). The thick contour line at 1.2
eV separates the region where the contour spacing is 0.3 eV from the region where
the contour spacing is 0.5 eV. There is a contour line corresponding to 0.12 eV added
to the MRCI(Q) PES to emphasize the depth of the product channel.

117

0.8
0.3
—0.2
—0.7

Energy (eV)

 

-2.7

 

—3.2

 

Energy (eV)

     

 

I
1...:
u:

     

 

3
R 4 5 4 3
“(1201):; s 7 133:1?0’0”)

Figure 8. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 135° BeFH system,
as described by the cc—pVTZ basis set, on the H~F and Be—F internuclear separations.

118

R ,H. (bohr)

 

4 5 6
RMF (bohr) RMF (bohr)

(e)

R H (bohr)

 

2 3 4 5 6 7 8
R1,“.(bohr)

Figure 9. Contour plots of the ground-state PES for the BeFH system, at 0 = 90°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (RBe_p = 50.0 bohr and 123.}: = 1.7325 bohr). The thick contour line at 0.8 eV
separates the region where the contour spacing is 0.2 eV from the region where the
contour spacing is 0.5 eV. There are extra contour lines corresponding to 3, 0.23, and
0.09 eV added to the CCSD(T), CR—CCSD(T), and MRCI(Q) PESs, respectively, to
emphasize important features of the PESs.

119

0.9
0.3
—0.3
——0.9
—1.5

Energy (eV)

—2.1
—2.7

 

-3.3

 

—3.9

 

   

 

  

 

1‘
A 0‘
>
3 _1.
E .
o "2‘
= .
L1: _3.
-4‘___.-
2
R1!

 

4 5 6 5 4 3 “‘2
41115011197 3 8 7 1g se—F (bohrl

Figure 10. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 90° BeFH system,
as described by the cc-pVTZ basis set, on the H—F and Be—F internuclear separations.

120

R I” (bohr)

 

23456782345678
RB”. (bohr) RIM. (bohr)

(e)

R H (bohr)

 

2 3 4 5 6 7 8
RBH.(bohr)

Figure 11. Contour plots of the ground-state PES for the BeFH system, at 0 = 80°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (RBe_p = 50.0 bohr and R34: = 1.7325 bohr). The thick contour line at 0.6
eV separates the region where the contour spacing is 0.3 eV from the region where
the contour spacing is 0.5 eV. There is a contour line corresponding to 0.1 eV added
to the CR—CCSD(T) PES to emphasize the depth of the product channel.

121

—0.3

Energy (eV)

—2.7

 

—3.3

 

—3.9

 

9

I
.—
I

l
'7’

_34

Energy (eV)

   

_4.
—4.6 ..... .

R11

4 5 6 7
‘1'”(bobr) 8

 

  

 

m’

l
9’
\O

7 6 5 4h31‘)
Rae—F0“)

Figure 12. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 80° BeFH system,
as described by the cc-pVTZ basis set, on the H—F and Be—F internuclear separations.

122

R l” (bohr)

 

23456782345678
RM. (bohr) R1,”.(bohr)

(C)

R H (bohr)

 

2 3 4 5 6 7 8
waohr)

Figure 13. Contour plots of the ground-state PES for the BeFH system, at 9 = 70°,
resulting from the (a) CCSD(T), (b) CR-CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (RBe_F = 50.0 bohr and Rg_p = 1.7325 bohr). The thick contour line at 0.6

eV separates the region where the contour spacing is 0.3 eV from the region where
the contour spacing is 0.5 eV.

123

Energy (eV)

 

 

 

—l.8
’3 V —2.5

'4‘ (b) 1 .. ~ -3.2

-5‘

’ —3.9
\ I 2
~4.6

4 5 6 5 4 3
(b01107 8 8 7 R6an (baht)

Energy (eV)

 

 

 

Figure 14. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 70° BeFH system,
as described by the cc-pVTZ basis set, on the H—F and Be—F internuclear separations.

124

 

R I” (bohr)

 

 

 

 

 

    

R 11-1: (bohr)

 

 

2 3 4 5 6 7 8
RM (bohr)

Figure 15. Contour plots of the ground-state PES for the BeFH system, at 0 = 45°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVTZ basis set. All energies are reported relative to the Be + HF reac-
tants (1236-1: = 50.0 bohr and R34: = 1.7325 bohr). The thick contour line at 1.3

eV separates the region where the contour spacing is 0.4 eV from the region where
the contour spacing is 0.5 eV.

125

Energy (eV)

 

 

 

 

Energy (eV)

 

 

 

 

Figure 16. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 45° BeFH system,
as described by the cc-pVTZ basis set, on the H—F and Be—F internuclear separations.

126

 

 

R1,”, (bohr)

 

 

 

 

 

2 ~ 3 4 5
RBH.(bohr)

Figure 17. Contour plots of the ground-state PES for the BeFH system, at 0 = 0°,
corresponding to Be atom located between H and F, resulting from the (a) CCSD(T),
(b) CR—CCSD(T), and (c) MRCI(Q) calculations with the cc-pVTZ basis set. All
energies are reported relative to the Be + HF reactants (R3941 = 50.0 bohr and
R34: = 1.7325 bohr). A contour spacing of 0.4 eV is used throughout the plots.

127

—0.1

Energy (eV)

 

 

Energy (eV)

      

 

 

3 .
5R3:F(bth)

Figure 18. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR—CCSD(T) and MRCI(Q) energies for the 0 = 0°, corresponding
to Be atom located between H and F, BeFH system, as described by the cc-pVTZ
basis set, on the Be—H and Be—F internuclear separations.

128

 

R H—F (bOhr)

 

 

23456782345678
Rkfl(bohr) RMH(bohr)

R H (bohr)

 

2

3 4 5 6 7 8
Rkn(bohr)

Figure 19. Contour plots of the ground-state PES for the BeFH system, at 9 = 0°,
corresponding to H atom located between Be and F, resulting from the (a) CCSD(T),
(b) CR—CCSD(T), and (c) MRCI(Q) calculations with the cc-pVTZ basis set. All
energies are reported relative to the Be + HF reactants (RBe_p = 50.0 bohr and
RH_p = 1.7325 bohr). A contour spacing of 0.3 eV is used throughout the plots.
The thick contour line corresponding to 3 eV is added to the PESs to emphasize
the presence of an artificially low and well pronounced barrier on the CCSD(T) PES
where none is present.

129

Energy (eV)

 

 

Energy (eV)

l
2'?"
m

 

 

Figure 20. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR-CCSD(T) and MRCI(Q) energies for the 0 2 0°, corresponding
to H atom located between Be and F ,BeFH system, as described by the cc-pVTZ
basis set, on the Be—H and H—F internuclear separations.

130

 

 

   

 

 

 

 

 

0.13 . . . 1 a
V
. \
\
K
s .
S
>
3
E
Lfl
<1
0.
-
CCSD(T)
CR-CCSD(T)
—0.06 l I l

 

 

45 70 80 90 l 135 I 180
0 (degree)

Figure 21. The dependence of the differences between the MRCI(Q) saddle point
energies and the (solid bars) CCSD(T) saddle point energies, and the (half-filled
bars) CR-CCSD(T) saddle point energies for the BeFH system, on the angle 0, as
described by the cc-pVTZ basis set.

131

R H (bohr)

 

2 3 4 5 6 7 8 2 3 4 5 6 7 8
waohr) RBH.(bohr)

(C)

R H (bohr)

 

2 3 4 5 6 7 8
RBH.(bohr)

Figure 22. Contour plots of the ground-state PES for the BeFH system, at 6 = 180°,
resulting from the (a) CCSD(T), (b) CR—CCSD(T), and (c) MRCI(Q) calculations
with the cc-pVQZ basis set. All energies are reported relative to the Be + HF
reactants (R364: = 50.0 bohr and RH_F = 1.7325 bohr). The thick contour line
at 1.3 eV separates the region where the contour spacing is 0.3 eV from the region
where the contour spacing is 0.5 eV. There is a contour line corresponding to 0.099
eV added to the MRCI(Q) PES to emphasize the depth of the product channel.

132

0.7

 

 

 

 

 

0.2

9 —0.3
G)

v —O.8

9 -13
2

1 -1.8

W :.

“-2.3

1 -2.8

"21 -3.3
A
>
d)
v
E?)
G)
:1
m

 

 

 

 

 

Figure 23. The dependence of the differences between the (a) CCSD(T) and MRCI(Q)
energies, and (b) CR-CCSD(T) and MRCI(Q) energies for the 0 = 180° BeFH system,
as described by the cc-pVQZ basis set, on the H-F and Be—F internuclear separations.

133

Figure 24. The possible mechanisms for the Cope rearrangement of 1,5-hexadiene.

 

m

0
Cyclohexane-1 ,4-diyl

B
R

Aromatic TS

v 4 .0“
\.

Bis-allyl

134

 

9M

 

 

// \

System I

H
C

 

//

System 4

 

NH

 

 

CH

System 5

System 6

 

 

Figure 25. The Bergman cyclization reactions.

135

55.0 '

“

 

H CASSCF (6,6)
—— UB3LYP (a)
50.0 H cosom

G—e CR-CCSD(T)
..... MCQDPT (6,6) I,
I

    

45.0

I!

40.0 -

AE (kcaI/mol)

35.0 -

 

 

30.0

 

1.5 2.5

60.0

 

H CASSCF (6,6) .
UB3LYP (b) .

 

55.0 “

a—a CCSD(T)
G—e CIR—CCSD(T)
------ MCQDPT (6,6)

I

50.0

 

45.0 "

40.0 ~

AE (kcal/mol)

 

35.0 r

30.0 r

 

 

 

 

5 01.5 1.7 1.9 2.1 2.3 2.5
R (angstroem)

Figure 26. The Cg}, cut of the PES of Cope rearrangement of 1,5-hexadiene as calcu-
lated with the UB3LYP, CASSCF, MCQDPT, CCSD(T), and CR—CCSD(T) methods
with the (a) 6-31G* and (b) 6-311G** basis sets. The energy, AE, is the energy rela-
tive to the 1,5-hexadiene reactant molecule. The solid squares and circles correspond
to minima on the CCSD(T) and CR-CCSD(T) curves, respectively.

136

 

 

 

 

1.63 r

1.53 0—0 D for the C21‘ PES cut 4
------ D for the 1,5-hexadiene reactant
OCR—CCSD(T) transition state

1.53 -

1.48 r

1.43 ~

1.38 -

 

 

 

133 n 1 A 1 m i A 1 A 1 A l A
I
1 1 . 1 1 1 1 1 1 1 1 .

H 7bu2 -> 8302
13—51 5a,,2 —> 5b:

0.56

0.3 .

N

0.2 ~

0.1 ~

 

1.5 1:7 A 1:9 ‘ 21 A 2:3 A 2:5 T27 L 2.9
R (angstroem)

 

 

Figure 27. The CR—CCSD(T) denominator Dm (top panel) and the absolute values
of the doubly excited cluster amplitudes t obtained in the CCSD calculations corre-
sponding to the 7b?l ——> 8a: and Sufi —> 5b: excitations (bottom panel), obtained with
the 6-31G* basis set, as a function of the interallylic distance R.

137

90.0

 

 

 

 

 

 

 

 

 

 

 

 

 

,HCCSD , 1
--—--—----- CCSD(T) », ’
8°") ” CR_—CCSD(T) G *
---— CR-CCSD(T)/D=1.0 ,
=3 70,0 _ ----- CR-CCSD(T)/D=1.4 ’ -
E . ------ MCQDPT (6,6) o ’,,-v """"" ~1
> o I"' ‘1, ————— ~
8 60.0 o ,.:// ~4-
i‘, // """""""
x, I "o‘
LLI 1 ’/ ,""‘
< 50.0 ‘\ I, I” ”’a
\\‘ \\ v I ””,’
40.0 \\ ”” ’I” ‘
“\—--— ’ov”” (a)
30.0 ‘1‘““7' 1 1
1.5 2.0 2.5 3.0 3.5 4.0
R (angstroem)
85.0 _o—eccso ,, .
_ -——--- Ccsom c
CR-CCSD(T) ,,
75-0 “ —--- CR-CCSD(T)/D=1.0 ,/ “
{g ’ ----- CR—CCSD(T)/D=1.4 ",yv """" ‘1
E 65.0 .. ------ MCQDPT (6,6) ” ,z", ------ ~.:
% I I'I"”’
I, ' I
g 55.0 , .43” ............
v \ ’ ’I vvvvv 1
LLI /’ ”’ ’v”’
<1 45.0 .- . ,1 xx -
. \ 2 I I
, \ : ’/’ ’I”
35 o ‘S"--”” x”
. ~ \‘\ x”, (b) 3
25.0 1 1 - 1 - 1 1 1
1.5 2.0 25 3.5 4.0

. 3.0
R (angstroem)

Figure 28. A comparison of the 02;, cuts of the PES of the Cope rearrangement of 1,5-
hexadiene, as calculated with the MCQDPT, CCSD, CCSD(T), and CR—CCSD(T)
methods, and (a) 6-31G* and (b) 6-311G** basis sets. The CR—CCSD(T)/D(T) = 1.0
and CR-CCSD(T)/D(T) = 1.4 curves were obtained by fixing the denominator Dm
in the CR-CCSD(T) calculations at 1.0 and 1,4, respectively.

138

 

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139

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